He is correct
Basically the way it works now is that whichever National Aviation Authority (NAA) (in this case Maxgolstreet wants a UK CAA issued EASA licence) you want your actual licence to be issued by has to be the one who issued you a medical.
If I understand correctly, Maxgolstreet, you have a EASA medical already issued by another European state other than the UK, correct?
That being the case if you did your training, lets say in the UK and then applied for your licence, it would have to be issued by whichever European authority issued your Medical (I think you can get it transferred but is a long drawn out process!).
If your intention is to hold a UK CAA issued EASA licence, then you are correct, you will need to have the UK CAA issue you your medical.
As for your original question of can you continue training with an EASA medical, the answer is yes (though I'm not sure which country you are referring to in doing your training!), in fact if all of your training consists of dual flights with an instructor, you do not even need a valid medial to train, nor do you need a valid medical to sit a flight test, the only time you will need it to be valid is for licence issue and then to exercise your privileges of the licence.
In terms of how long it takes, I can't help you there, I suggest a quick call or e-mail to the UK CAA would help you with that.
Good luck.