Some outdated and incorrect information.
The CAA closed its Aeromedical Centre - the bit where medicals were done, after a consultation. You can find the details on the CAA website under 'closed consultations'
There are currently 3 AeMCs in the UK where you can do your initial Class 1 Medical, one is in Gatwick, the other in London and the third is at Heathrow.
I don't know why there are none in Birmingham, Manchester or elsewhere in the UK but do know that the market for initial Class 1s, is limited and that a large number of initial Class 1s is for people who are not UK based and therefore being close to large transport hubs may be why these centres are based may be not a bad idea. Setting up an AeMC is not trivial, both in effort and costs.
EEGs and Chest X-rays have not been part of the initial Class 1 for many many years.
NATS has not been doing the initial Class 1 medicals for quite some years. That is not because they have been de-authorised (they still do initial EU Class 3 medicals for ATCOs which requires the same authorisation) but because they have a capacity issue - as I understand it.