Odd connection loss problem limited to certain websites

[#1]

Hello there,
There's no problem with the speed of the connection. My problem is that certain websites (including major ones, such as www.google.com, www.bbc.co.uk/news, www.facebook.com, www.plus.net, etc.) don't allow me to connect at certain times, very regularly. I can temporarily fix the problem by disconnecting from my WiFi network before reconnecting, but then within 5 minutes or so I have intermittent access again to certain websites. This pretty much affects all websites, but only a certain subset of them at any given time.
When I 'ping' these websites while I can't connect to them, I indeed immediately get a 'Request timed out' message. If I refresh the website a minute later it's usually available again.
I tried to use Plusnet's Broadband diagnostics tool, although their downloadable one doesn't support Windows 8 (?!) and their online Broadband Troubleshooter gives me "Error" every time.
I've tried resetting the router (using the reset button), but to no avail. My laptop is brand new, purchased within the last week.
Many thanks,
Jamie

[#2]

And just to give some idea of the severity of the problem, it's roughly 50% of websites I can't access at any given time (e.g. my bank account just now!)

[#3]

Which DNS servers are you using on your pc and laptop?  Plusnets have been problematic for a while, so its always best to have OpenDNS servers or Googles as a secondary as a backup.
google 8.8.4.4
openDNS 208.67.220.220

[#4]

Have you tried finding the WiFi channel (1 to 13 in your router) that gives the most reliable wireless connection ?
If you are using a WiFi channel that gets interference from your neighbours, you might see good WiFi signal strength,  but the connection reliability can be poor.
Initially try WiFi channels  1, 6, and 11,  and see if using any of those makes the problem disappear.

[#5]

From the symptoms it sounds like a DNS issue rather than a connectivity/wireless issue. For example, I can watch a full program on iPlayer without any interruption (since no host needs to subsequently be resolved after I've initially connected to the site). And as I've said, at any given moment I can always access at least some websites!
I'm not techie enough to start fiddling with DNS settings though. Will do some investigating and then phone PlusNet if I can't fix it.

[#6]

Here's a simple test:
On your PC, go to Start > Run > cmd (hit return) > nslookup www.google.com ;   (and try the other sites)