Hi Kazi
Sorry to hear you are having problems - I moved from BT to o2 this year and have been on - but then I'm only 2km from the exchange.
As someone else suggested you need to check the stats on the router to see if it's syncing ok, I don't use the Thomson o2 supply so I haven't seen what it has - I suspect it'll be minimal but it might give an idea. You need to see if the problem is the connection or not.
You don't say if you've gone through the usual checks so.... as a test I always recommend removing everything else from the phone system to test - if you have a "proper" BT master socket take out the lower panel and plug the microfilter in the hidden socket - this should cut out all your internal wiring, sometimes that and the phines can cause problems.
If this isn't possible just unplug everything except the router (make sure you remember sky boxes and fax machines (faxes cause me the greatest issues with BB) then reboot the router and see what you get. As you have such a low speed you should see if there's a difference straight away. If there is it's a process of elimination - connecting stuff until you find a fault.... last one of those I saw was a dodgy skybox - worked fine - would dial up etc but put a nasty hum on the lne when connected, so I disconnected it. Whn Sky rang the customer to say "we can't see the box anymore" she told them her BB supplier had condemned it and they tried to charge for a repair! One other thing is to change the microfilter - they DO go faulty and some are better than others (the BT ones are expensive but good - the cheapies can be just as good but quality varies so it's worth a swap or two)
Anyway - if problem is stil there then it's unlikely to be your bit. BT can test the line for simple faults but sometimes it's just the quality of the copper (or sometimes aluminium) which causes a problem - BT should be capable of finding it but will be reluctant unless pushed by o2. Have you tried picking up the phone and dialling 1 then listening - it should be pretty much silent - if there are hums or crackles they are going to induce errors, mention them to o2,BT Ask for a pair change.
If the line speed is OK there may be other issues with the PC or the network connection - I doubt that though as you seem to have done those tests and anyway they shouldn't drop it so far. One final tip is to threaten to change providers as they're not giving you adequate service, trouble with that is most ISP's use BT anyway and if it's the line......
Hope this is of some use.