To clarify:
With FTTC (Infinity 1 or Infinity 2 in BT Retail speak), you will still have a copper pair all the way back to your local exchange, for voice. As BT Wholesale do not offer any options that do not include voice capability, thats non negotiable, no matter which ISP.
Also with FTTC, there will be a fibre cable from the new cabinet to the serving exchange, which may or may not be your local exchange - mine is 12 miles away.
The copper voice and the fibre data are joined at the cabinet, and sent down the copper to your home.
Its unusual to remove the old cabinet though, that normally stays put, as it holds your E and D sides, with a tie cable to the new cab for joining into the mini DSLAM.
FTTP does not offer voice, so a fibre from your serving exchange straight to home, spliced as near as possible to your home. This is expected to be offered to all FTTC in the not too distant future, with an install cost around £1k