Yes, you can hold both. The UK CAA is now completely separate from EASA. Comparable to EASA vs FAA.
At this time, there are no agreements or associated Annexes in place between the EU and the UK in this area. Licensing and Training under the oversight of the UK CAA is now considered as third country licensing and training
Irish Aviation Authority - Brexit Notice
- From EASA to a UK application: possible.
- From UK to an EASA application: not possible. (Sadly EASA have "shut the door", it is very unfortunate)
If you hold an EASA licence you can apply to the UK to hold an equivalent UK Part-FCL licence. It involves the inevitable administration costs from both the UK CAA and the EASA state to send details. You will also require a UK CAA Class 1 medical - initial/revalidation at the discretion of the UK CAA AME (not the UK CAA itself).
If you are a former UK Licence holder, there are some further considerations.
UK CAA - Eligibility for application
Licence holders who fit the below criteria can apply for a UK PART-FCL licence > using this application route:
pilots, instructors and examiners who previously held a UK issued Part-FCL/BFCL/SFCL and have not had a licence issued at a higher level since 1 January 2021
EASA licence holders whose licence date of initial licence issue is prior to 31 December 2020
Licence holders who fall into the below criteria cannot use this route and must instead email FCLweb@caa.co.uk (subject heading ‘Attention Tech: EU Conversion’) for further advice:
pilots, instructors and examiners who previously held a licence with the UK CAA and have had an EASA licence issued at a higher level since 1 January 2021
EASA licence holders whose licence was issued after 31 December 2020
UK CAA - Conversion of an EASA flight crew licence to a UK part equivalent licence