GMT is the correct time zone
For London, yes. Lands End is 5.7 degrees west, Peterhead is 1.7 degrees East. There's 30 minutes of difference in sunrise straight away (29 minutes 35.5 seconds, for pedants) even before you consider the effect of Scotland being substantially further north.
And if we're being super-pedantic, GMT (which is equivalent to UT1) is not maintained and there is no accurate source of it that is generally available. Everything, including the "Rugby" time signal (now actually broadcast from Cumbria), the "Greenwich" time signal on Radio 4, beep beep beep beep beep beeeeep, GPS, whatever, is actually UTC. There was a bill in parliament in 1996, which fell with the election of the Blair government and hasn't been looked at since, to change UK legal time (as defined by the Interpretation Act 1978) from GMT/UT1 to UTC. At the moment, the difference is less than a second, but this is going to become a real issue if (as seems quite likely) leap seconds cease to be inserted into UTC.