My question is, do BT have to be the ones to reconnect?
No, not any more.
In a deal with the telecoms regulator Ofcom, BT created the separate Openreach division in 2006, to which all responsibility for outside lines was transferred. Although a subsidiary of BT, Openreach is obliged to install and maintain outside line plant on the same terms to all service providers, including BT itself (the charges are actually set by Ofcom).
Any of the alternate providers can, therefore, arrange to have a new line installed (or an old line reconnected) for you via Openreach. Some offer this service, others still insist that you must have an operating line with BT first, but that's entirely that provider's specific policy.
It's worth making inquiries to all the providers you might consider and to see if they'll "do a deal" on installation fees. These days many of these charges are open to adjustment if you agree to a longer minimum contract period, subscribe to extra services, etc.
The line leading up to the house is BT's responsibility (once it's inside, I'm uncertain).
BT/Openreach's responsibility ends at the NTE5 where the outside line first terminates inside the house. The NTE5 is the unit used on all new installs which has a removable panel to give you access to a test jack, like this:
Extension wiring from there to any extra jacks around the house is your responsibility.
Openreach will install extra jacks for you, but you'll find it much cheaper to find a local telecoms engineer to do it, if you don't feel able to do it yourself.