The issue is rather more immediate than just the Class 1 medical. Since the UK CAA issued your LAPL and, presumably, currently hold your medical records, the application for the PPL will have to be made to them and not the DCA, and you will need at least a Class 2 medical to apply. Application for the licence can be supported by a medical certificate issued outside the UK but there are certain hoops that you will have to jump through. Have a look at the paragraph titled 'UK EU licences'
here.
I think you may have misread the CAA website as I cannot find anything to suggest that you must have a medical issued by the same state that issues the licence. You can gain or renew an EASA medical certificate in any member state but the details must be sent by the AME to the state that holds your medical records. The exception to this is Germany where their national legislation data protection legislation does not permit the transfer of pilot medical records.