In terms of dealing with BT they will only do something if you can prove that the speed you were recieving previously was greater than you are currently recieving. If it has always been slow they won't determine there to be a fault.
They can do all sorts, remotely, to determine if there is a fault on the line so ensure they have examined that route prior to sending an engineer, as in my experience an engineer very rarely resolves an issue.
Also remember that BT Broadband and BT Telephony are 2 seperate companies so it can be abit of a pain pinning a fault on a certain 'division'.
Up until you can prove something is actually wrong with the line, you can only try simple things like powering off the router for 5 minutes then turning it back on afterwards, then booting up your computer.
Deleting temp internet files etc can also help.
Easiest thing to do would be try Firefox as mentioned above, but it's unlikely to fully resolve your issue.
Tim