If you are using DHCP on the BT router to allocate your device addresses, rather than static configuration, the BT router will have a record of the IP addresses that it has allocated to your devices using DHCP. When you remove the BT router, if the physical links to your devices stay up then the devices will retain those IP addresses. When half the DHCP lease time expires, the devices issues a DHCP refresh request which allows it to retain the same address.
When you then install the backup router, it has no knowledge of what IP addresses are in use by those devices, until it receives a refresh request from the device. At that point the new router will become aware of them and record the allocations in its DCHP table, if the address is available or refuse it, at which point the device will request a new allocation. There is a window within which if a brand new device comes online it could be allocated the same IP address as an existing device of which the new router has yet to become aware.
If you are using a a separate device as the DHCP server, or using static addresses, then there is no problem.
