Logbook Experiance
 

Can I log my experience in a CAA UK logbook and apply at a different authority for the licence, or should the experience be logged in the appropriate authorities logbook.

I do not see why you should not use the CAA logbook. Part 66 requires that you record experience and does not specify a particular log book. You should refer to Part 66 and Annex I & III do identify the extent of experience that is required

AMC 66.A.10
1. Maintenance experience should be written up in a manner that the reader has a reasonable understanding of where, when and what maintenance constitutes the experience. A task-by-task account is not necessary but at the same time a bland statement “X years maintenance experience completed” is not acceptable. A logbook of maintenance experience is desirable and some competent authorities may require such a logbook to be kept. It is acceptable to cross-refer in the EASA Form 19 to other documents containing information on maintenance.