From your line with BT Total
Download speedachieved during the test was - 1857 Kbps
IP Profile for your line is - 2042 Kbps
Unless you can improve the connection speed no other ISP is going to go much faster, or are you saying that speedtests at peak times are a lot worse than this with BT? Alas as peak times coincide with when its dark, it is entirely feasible the bad performance is due to the wiring issues, rather than the ISP. Have fixed a few lines for neighbours where once working cleanly they were surprised how much better browsing felt, and not just getting faster speed test results.
From list of ISP's
Virgin over a telephone line - avoid like plague
Sky - if their LLU you MUST take their telephone line rental now, i.e. full LLU, but service is good
TalkTalk bad reputation for phone support, and full LLU to, but does work pretty well.
BT to be honest, every one hates them, but on the line I have access to it works as well as any other.
There are 100's of other providers that could be mentioned, they might cost more, but offer different things.