Brocket, Openreach will provide FTTP, in your case it can be fed overhead the same as your present line is. If the pole at the end of your lane, or one close by, has what is known as a Connectorised Block Terminal (CBT) & from what you say it should have, then fibre cable has been provided to that, so, a bit more fibre cable fed overhead the same way as your present installation will get you FTTP. There is a limit to the length of the span, between poles though & it may be less than with a copper drop wire, I don't know for certain though.
For those with copper cable & slower speeds than you should be getting this can be for all sorts of reasons & Mr. Nick seems to have had a few of them! Poor/corroded connections & damp are a few but using aluminium was the worst thing BT ever came up with (well, maybe not, but certainly one of them) as it's conductivity is almost half that of copper, it's weak & brakes easier but, I presume, at the time it was a cheaper alternative?
The other thing that is important to give the best speed possible is for the external (copper hopefully) to feed directly into your 'master socket' and your router to be plugged directly into that. If you have multiple sockets in the property they need to be fed from the connections in the cover plate of the master socket. If you can do without those sockets, let's say they've been in for years & you use cordless phones or as in Brockets case don't have a land line, remove the sockets or at least disconnect the wiring from the master.
Having said that, I'm guessing I've never looked, your contract will say "UP to X mbps speeds", not minimum of.