In the UK most of the physical infrastructure is owned by OpenReach which is a separate company distinct from BT (Retail) who sell voice and broadband services. The OpenReach/BT separation was done to stop BT getting a competitive advantage over other ISP's. The "rules" are such that OpenReach have to supply lines to all service providers for the same price.
As such, you don't buy "a line" from OpenReach directly, it's always provided "second hand" in that your service provider leases the infrastructure from OpenReach. But you can end up with the oddity that having bought a service from Colt, C&W, BT, Sky (whatever,) it's an Openreach engineer that turns up to install, terminate and "prove" the physical line. Albeit, that they won't (usually) be installing the ISP equipment - the "router" in SOHO use cases, that's provided by the ISP.
In short, (and somewhat inaccurately,) OpenReach provide the wires, the ISP provides the service and the equipment. (Though doubtless there's plenty of "rules" imposed by OpenReach about what can and can't be drive down "their" wires.)
By way of made up example, it's highly unlikely that Sky will be prepared to give you a service using an incumbent router that belonged to (let's say) PlusNet. Sky will want you to use a Sky router because they have to support it.
We haven't discussed why it is you are determined to use the incumbent kit. Do you think, for example, it's going to be cheaper (I doubt it) or is it going to cause some crime against the decor...? If not, I just wouldn't worry about it and go with whatever your new ISP supplies.