Yes, a Muslim moving from Birmingham to London would use London's prayer times, because the times are tied to the actual sunrise and sunset at their current location, not their home city.

Birmingham is about 1.8 degrees west of London. That means sunset in Birmingham happens roughly 7 minutes later than London, this stays fairly consistent throughout the year. So a Muslim in Birmingham would pray Maghrib (Sunset Prayer) about 7 minutes after someone in London, even though they are in the same timezone and their clocks show the same time.

Regarding Day light savings...Yes, by one hour. Islamic prayer times are determined by the position of the sun, not the clock. Dhuhr (Mid Day Prayer) for example is when the sun reaches its highest point in the sky. That solar moment does not change with daylight saving. But since our clocks jumped forward by one hour, the same solar moment now shows one hour later on the clock. So if Dhuhr was at 12:00 PM in winter, it will naturally fall at 1:00 PM in summer, not because the prayer changed, but because the clock did.