Actually the full CAA statement clearly states that it is talking about "use on UK-registered aircraft" - and it says absolutely nothing about F or D-reg or any other reg. In fact I am struggling to see anything that is clearly a falsehood in the statement - per
The UK CAA would NOT be able to make any comment about flying a F- or D- or OE- reg aircraft as it would be outside their juristiction. If, and that’s a big if, the UK were not be able to remain within EASA, any EASA UK licence would become automatically a national ICAO compliant one. To fly an EASA reg aircraft one would need either an EASA licence or a validation based on their national ICAO licence.
Hence why easyJet has blinked first and asked its EU based pilot to change SOLI from UK CAA to Austro Control.