There is a tiny possibility that the route back to the exchange from the new address could be much longer, let's hope not. Ideally you would see if you could get the line stats from the current occupant's router

.
Ignoring that, two things are preventing those high speeds where you are now. First you are almost certainly on an "up to 8Mbps" package, whereas nearly every ISP on that exchange offers "up to 20 or 24Mbps".
But secondly, despite that, you should be connecting at 8Mbps. The drop from that could be down to the house phone wiring, which you can ignore where you are but need to check for at the new property, or to something like an unfiltered Sky box or other problem. So either issue could go with you and severely affect the speed you get compared to the expected.
Sky LLU has been mentioned. Since 1 September that is
Full LLU only, which can be a pain if you want to migrate. The service itself should be fine on the unlimited package.
TalkTalk - many of us are against it, and your usage makes me think you would get considerable traffic management effects.
Orange has been mentioned. When that was first transferred to BT Wholesale
White Label it seemed an improvement on Orange LLU, but there seem to be a few less happy punters turning up more recently.
To what extent is price an issue? You can get an LLU package using the TalkTalk exchange equipment, (but not their network), with unlimited allowance and no traffic management, but you would be looking at £40pm or so

.
On "normal" BT Wholesale based connections, you couldn't go far wrong with
IDNet Home Pro as long as you can keep the peak time usage within the 40GB. The allowances are applicable at the weekend as well though.
There is also BT Broadband itself of course, which is unlimited but has traffic management, and even Plusnet is worth looking at if you can stay withing the 80GB peak usage, which IIRC is weekdays only. That has a "Pro" option to remove the traffic management and priorise gaming, if that is of interest.
BT Wholesale products are not as bad as they were, as the new WBC DLM Profiling system is much better than the old one and no longer messes people up as much.
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