It depends on how important it is to you to have minimal downtime AND retain the number. Both were for me.
My solution was to order a new FTTC/SOGEA line (no FTTP here) with a new master socket in a better location, with PlusNet.
Once that was up and running I cancelled my old ADSL (NOT the phone) asked A&A to port the landline number to them, it took a few days.
The port terminated my old BT line, though it still has dial tone and a different number allocated (according to 17070).
It was all somewhat easier for me as I had separate contracts, both on "out of contract, minimal notice" terms with Plusnet (ADSL) and BT (POTS).
Both PN and BT actually refunded me overpayments back to the date of termination/transfer.
The extra cost of installing a new line was soon recovered by the monthly savings.
Also you don't actually say who your landline is contracted with (BT may "own" it but is it BT retail you pay?) and whether it has broadband over/associated with it and who supplies that?