Where do I go from here?

[#1]

No, Stallan not silly I'm sure.  I've just done the switch over but my wife will not be too pleased since she has to now step over a Thomson & cables to get to her desk top!  Have to put a red flag out as a warning.
Thank you for your interest - much appreciated.  At this stage anything is worth investigating.

[#2]

Those results are very informative, especially the noise margin plot. From 11:20 to 11:45am (25 minute period) your connection was subject to very high interference. Had the ongoing sync been done at a 'good' (low noise) time, the sync speed would probably have been much higher with the noise margin at 15dB. That burst of noise would almost certainly have caused sync to be lost with it being regained at very low speed - sustaining the awful IP Profile you currently have. So identifying what caused that noise seems to be the key.
Stallan has suggested one possibility, since this was a wireless connection to the router it could be Wi-Fi interference, internal to your property or a neighbour, e.g. an electrical appliance (a keep-fit treadmill for example, work bench, kitchen appliance, …). Changing channels can sometimes help though I suspect that noise would swamp all of them. Finding out if the noise still occurs when using a wired connection is useful, and I suggest also turning wireless off on the Thomson (click Home Network->Interfaces->WLAN->Configure (top right) untick the 'Interface Enabled' box. Testing with the router plugged into the test socket (using a filter) is desirable to reduce risk of an internal cause.
Externally does your telephone wire run overhead? Connection box full of water? Close to trees? Past a factory? Anything that might generate electrical noise? It's possible a faulty cable joint is the cause though I'm unsure if the noise history shown is consistent with such a source.
Kitz's lowSNR page is worth reading, and especially the link to REIN (Random Electrical Impulse Noise). You might need a visit from a BT REIN specialist, but I expect more investigation will be needed before that stage is reached.

[#3]

A good answer from spraxyt @12:28, followed by some pretty good graphs for the noise margin.
The second answer from spraxyt is also good. I too noted that the kind of disturbance shown would have knocked any normal ADSL over.
Your connection survived at the time because it was working with an SNR of over 26db.
I'm pretty sure that, if you listened to the line at that point, you'd have heard something - so if you happen to see this kind of behaviour again, go pick up the phone.
If you can hear something then, that might be the time to phone BT (154?) to report a fault on the line - then they will be able to hear it too.
Report it as a voice fault, with no mention of ADSL, broadband or computers, as that will sidetrack things.
Also mention that it is intermittent. Unfortunately that makes it more of a ^%$^ to identify, and more of a problem for BT (well, probably BT) to fix.
The more evidence you gather, the better...
    Mike

[#4]

I also wanted to say that the spike at 12:56 is harder to determine. There doesn't seem to be any onset in noise that has caused a disconnect.
In fact,  there is probably only a spike on the noise graph *because* the connection has dropped, rather than the other way around.
Does routerstats also give you access to any of the error statistics for your line, such as FEC, HEC and especially the CRC and ES figures?
If it does, it might be worth graphing those too, to see if you get anything shown there that isn't showing up on the noise-margin graph.
Mike

[#5]

I don't think the 'drop' at 12:56 is significant. Routerstats gets its information by telnet into the router, and that single interrogation will have failed (perhaps because the router was busy on other tasks). The information is shown as 'zero' if not collected.
Routerstats Lite can only do noise margin and sync speed, but the full product can access the error information pages (and bit loading too), so potentially graph them. The full product works "out of the box" with the Thomson. I had thought of suggesting downloading that but thought it might be adding unnecessary complication. I imagine the error count shot up during the burst of noise.

[#6]

Argh - hate that. If the collection program fails to take a reading, it needs to distinguish that from a zero reading, which can look like a significant problem.
I agree that the earlier situation doesn't need anything fancier than the SNR measurement though. It's easy to spot, even if it won't be to fix.

[#7]

Spraxyt & Mike - much thanks for all the early morning information.
Will get down to the gritty stuff later & have disconnected wireless on Thomson.  Incidentally, I have an ADSL Nation on master socket. 
For now, just an update.  PN & their 'provider' did something [? ]yesterday I was informed.  What I do not know but noise seems to have settled a bit:-
Below some data as of 7.00am today.  I have not done a router restart, although tempted, to see if stats change.
Ian

[#8]

Well, just did a router  'restart'. 
Looking a healthier connection speed @ 4928 kbps & SNR now15.  Will assess situation on return home this afternoon; maybe the d'load speed will have risen from 95kbps early this morning.  Here's hoping.

[#9]

It's useful to have confirmation that your target noise margin is 15dB. The increased sync speed fits in with my expectation in the first paragraph of reply #40, quoted here for convenience:

Quote from: spraxyt

Had the ongoing sync been done at a 'good' (low noise) time, the sync speed would probably have been much higher with the noise margin at 15dB. That burst of noise would almost certainly have caused sync to be lost with it being regained at very low speed - sustaining the awful IP Profile you currently have. So identifying what caused that noise seems to be the key.

Last night's noise periods were less severe, and even with that level the router might well have stayed connected with noise margin falling from its elevated level. However this is still unsatisfactory. Interestingly there are a couple of bursts of noise either side of 3am, a time you mentioned yesterday. After the burst of noise around 5am noise margin settled to a slightly higher level and variations since then seem smoother. I wonder if that is when BT 'did something'? Hopeful, but as usual time will tell.
Some way to go yet but if the bursts of noise have been eliminated making the normal 6dB target noise margin viable your sync speed would increase to about 6.5Mbps which I think was around your original expectation.

[#10]

Just an update @ 11.30pm. 
RouterStats Lite shows a continuous 'unspiked' line from 5.00am today until now [I'll leave everything on o/night to plot progress etc].
SNR has dropped [was 26] slowly from 15 > 12.5 but IP profile still stuck @135 & d'loads 94kbps.

[#11]

Good news. Fingers crossed.   The 2½dB reduction in SNR is not unusual during evening hours.
Hopefully the DLM will relent and increase your IP Profile tomorrow morning, it can take three days but for this level of increase it should be much sooner.

[#12]

Saturday morning things are starting to happening but what?  Latest summary is [b'up stats below]:-
- two blips on SNR @ 1.50 & 3.14 am; the other two @ 7.20/7.40am are mine when I reset router & removed a wire!
- PN in a closed message on my open ticket @ 4.00am today state, 'bandwidth has been reset'.
- BT speedtester IP profile is now 3500 & I asked PN to match [helpfully done]
- Line d'load is not back up as yet  & hovering around 200kbps still.  OK better than the miserly 94kbps of yesterday
Q1. I wonder when I might expect a matching increase in d'load speed? Q2. could the two early morning blips be at the PN end switching something?

[#13]

Have you disconnected/reconnected to Plusnet since the CSC changed your PN Profile? That doesn't require a router restart (re-sync), click Broadband Connection in the router's left-hand menu, then click the Disconnect button (on the right). The button should change to a Connect button, wait a few seconds then click it again to re-connect. The should bring the PN Profile into operation.
You can verify the PN Profile in the Member Centre (Connection Settings->High-speed Broadband), shown as Current line speed. The speed you got on the BT test is consistent with the previous value being 250kbps which I think is the minimum PN setting even though you had 135kbps from BT.
As far as the disconnection around 1:50am, that might be self-induced by your router, the Thomson's do this sometimes. Could be BT doing work in the exchange (unconnected with your fault). It probably reduced your sync speed (to restore 15dB noise margin), but doesn't look to have had a bad effect. The noise just after 3am might be another instance of the '3am' mystery seen previously.

[#14]

Really appreciate your continued, reasoned thinking Spraxyt.  A great encouragement when faced with a pretty dismal situation.  Between you & WWWombat, your advice/tutorials have assisted. 
I now have a d'load speed bordering on 3Mbps.  Just hope PN will still beaver away with BTw behind the scenes and stay on top of it.
Incidentally, I came across this 12 mths old press release:
<http://www.btplc.com/news/articles/showarticle.cfm?articleid=CA7AABCD-8EE4-41DD-BE7A-C6A2B1EC3454&Terms=23> 
Many I speak to still query who owns what?
I will continue monitoring my situation for long term stability. 

[#15]

Glad to hear you are getting better download speed now. As you said demonstrating connection stability is the next hurdle. Your target noise margin should then reduce, and sync speed increase, but it can take a few weeks before the DLM accepts this. Restarting the training period on your line would hurry it along but that shouldn't be done until stability is assured. Are the occasional noisy incidents still happening (particularly the 3am one)? (The frequent 0.5dB up and down ones don't matter, that is normal.)
That press release you referred to concerns the BT Plusnet Partner managed broadband service which is offered as a white-label service to resellers and virtual ISPs. The name should not be read as BT_Plusnet, it is a joint venture between two separate companies. The BT partner is BT Wholesale, the division of BT which provides and operates their broadband infrastructure on a commercial basis. Plusnet is owned by BT Retail which provides ISP services just like Plusnet does purchasing services on the BT Wholesale infrastructure. Openreach is another division of BT which provides telephony and local loop services on a commercial basis.