BT will blame your pc, its a easy way for them to get out of doing anything.
with a ISP you really do have to stick to your guns or they will walk all over you.
if you can change your router to use open DNS servers that will see where the problem is.
if all sites now work its a BT problem and you should advise them that under open DNS the sites work, when i use your DNS server they don't work.
Open DNS servers are as follows
208.67.222.222 and 208.67.220.220.
if your unsure on open DNS have a read here, its a free service that i sometimes use myself when my ISP has DNS server problems.
http://www.opendns.com/if you read your router manual it will tell you how to change those settings.