If you'll all excuse what will turn out to be a rather verbose post, I'll see if I can explain the situation from the technical side of things in a little more detail.
Let's start with the basic setup, assuming that only BT is involved (as was the case some years ago, before other companies could offer line rental). With today's telephone system, when somebody moves out of a property and cancels telephone service, nothing needs to be physically disconnected, since the switching equipment is all remotely configurable in software. A command is simply sent to his local central office (telephone exchange) telling it to discontinue service on his line. At the same time, the line records are amended to show that every pair of wires used between the C.O. and the house is now spare. The wires linking the pole outside the house to the exchange aren't permanently fixed to that single route. There will be a huge cable carrying many hundreds of pairs which runs from the exchange to the first distribution point (those big green cabinets you see along the street), then smaller cables which fan out from there to more distribution points around that neighborhood, and so on, until eventually it gets down to the smallest roadside pedestals or pole-mounted junction boxes from which the individual drop cables connect to each house. At each of those distribution points, any pair from one cable may be cross-connected to any pair on another cable to get service to a particular address.
So the pair of wires to the house in question might, for example, be on pair #495 of the big cable to the first distribution point, then on pair #106 on a cable which runs from there to a green cabinet at the end of the street, then on pair #18 which runs along that street to a junction box atop a pole, then finally onto the drop wire which runs to the house. So when service is discontinued, pair #495 in the first cable, pair #106 in the second cable, and pair #18 in the third cable will all be marked as spare, but for the moment they will all remain connected together, still linking the telephone jack in the house back to the C.O.
Now, if you move into that house while that state of affairs subsists, then all that is necessary to start new service is to send the appropriate command via the software back to the switching equipment to tell it the new number and to start allowing incoming and outgoing calls.
The telephone network in a particular area is always changing as people move in and out, request new lines and terminate old service, etc. So if a new line is required somewhere, any section of the cable route which previously served the address in question is "fair game" to be reassigned to somebody else. So, to continue the example, if somebody else in the same street requests a new line, pair #18 in the cable which runs along your street might be disconnected at the pole outside your house and then jumpered to a drop wire to a nearby house. Or a new circuit might be needed between the first and second distribution points I used as examples above, so pair #106 in the cable which formerly served your house might be diverted at the second distribution point to connect to a cable which runs into some other nearby road. Basically, it all depends upon how much of the line capacity is already in use, how much demand for service there is the area, and so on, but the longer the house stays without service the greater the chance that at some point part or all of circuit it once used will be "stolen" and reassigned to some other line.
If you move into the house and request service after that has happened, then it will be necessary for an engineer to find whatever pairs of wires are spare at the time along the entire route and then visit one or more of those distribution points to make the appropriate cross-connections to get service back to your house. Depending upon local conditions, it might mean just climbing the pole outside your house, or going into the underground chamber by your front gate. But it might mean visiting five or six different locations between your house and the exchange to make all the connections. In some cases it might involve running a new cable along several spans of poles if there are no other circuits spare.
This is why BT will quote a standard reconnection charge for "new" service, even if there was a previously telephone service at the address in question, since until the order is put through to the local area nobody will know exactly how much work is entailed in re-establishing service. But as many of you have discovered, this tends to be negotiable at the billing/accounting level. Engineers generally aren't too worried about it, and certainly at this time of year you don't want to be climbing up poles and messing around in underground chambers any more than necessary!
O.K., now we get to LLU (Local Loop Unbundling) which is where it gets messy due to third parties being involved. All of the outside line plant still belongs to BT, although it has now been transferred to the still-fairly-new OpenReach division. BT/OpenReach is still responsible for installing, maintaining, and doing all that outside cross-connect work on the cables which run around town.
There are two ways in which LLU can take place. In most of the major cities and larger towns now, many alternate providers now rent space inside BT telephone exchanges where they install their own switching equipment. In these areas, when you get LLU service from somebody other than BT, your line is physically jumpered to that provider's equipment at the C.O., while BT/OpenReach is still responsible for doing all the outside line work which is necessary to get the line to your home. In places where LLU operators do not yet have their own equipment, service for your line is still provided by the BT switching equipment, and your chosen provider simply pays BT for that use out of the line rental you pay them.
But either way, the outside line plant is owned and maintained by BT/OpenReach, and under the applicable contract once a line has been transferred for all billing purposes to another provider, only that provider may then order any changes or cessation of service on the line. That's why things like reporting a fault then become a two-stage process where you notify your provider, then if he decides it's a physical fault on the outside line itself, he calls OpenReach to get the problem fixed.
Unfortunately, this is what gives rise to the problem of BT being unable to simply "take over" an existing service unless and until the current service provider relinquishes control, even when the occupant is vacating the premises and clearly will no longer need the line. In that respect, it is not entirely BT's fault, since as the incumbent telco those were the conditions imposed on it by governmental ruling. There certainly is room for a much smoother way of dealing with everything and for the average subscriber to find out who is providing what though.
DSL service adds another level of complexity to the problem, but that's another long post in itself.