It depends where the fault lies, if it is in BT's infrastructure, it will be the same speed no matter which provider you use. What will speed things up is if you have carried out all the diagnostics you can to rule out an issue on the premises causing the problem. It's worth rebooting the router/hub if that's not been done in along time, it may sync back up at higher speed.
i) They will probably ask you to plug the router/hub into the isolation test socket located behind the master socket (2 screws on the face plate of the socket need to be removed, behind it you will see a test phone socket).
ii) If you have analogue phone, you can plug the phone into the master socket and dial 17070, option 2. This is called a quiet line test. You will here an automated voice, you want to listen for any noise or crackling on the line. Noise can cause interference with the DSL and cause speed issues and rapid disconnects.
iii) They will ask if you performed the speed test using a wireless or wired connection, if wireless, repeat the speed test wired, it could be your wireless card/signal causing the issue.
iv) If you have a spare adsl filter, try that, it could be the filter is on its way out.
v) If you have a fax machine on the phone line also, remove that and test the speed again to see if the fax machine is causing slow speeds.
BT should have remotely tested the line using their remote diagnostics by now, they may see a fault straight away and you might not have to carry out any of the above.
As far as I am aware there is no SLA for ADSL, it's a best effort from BT. If you can rule out everything your side causing an issue, BT will have to send an engineer out to investigate.
HTH.