Agree with MJ.
As far as the N-reg is concerned - this is the result of FAR 61.3:
§ 61.3 Requirement for certificates, ratings, and authorizations.
(a) Pilot certificate. No person may serve as a required pilot flight crewmember of a civil aircraft of the United States, unless that person—
(1) Has a pilot certificate or special purpose pilot authorization issued under this part in that person's physical possession or readily accessible in the aircraft when exercising the privileges of that pilot certificate or authorization. However, when the aircraft is operated within a foreign country, a pilot license issued by that country may be used;
So the only reason you can fly an N-reg on a UK license within the UK is because the US government allows you to. (And note that the US doesn't understand the JAR-FCL and EASA thing. If your EASA license is issued by the UK authorities, you can only fly within the UK on an N-reg. But if that same EASA license is issued by the Dutch authorities, you can only fly within the Netherlands.)
The logging you just log what you have done.
Just remember that if you fly on a UK/JAR-FCL/EASA license, you have to log according to the UK/JAR-FCL/EASA ruleset, and if you fly on a US license, you have to log according to FAA ruleset. Which may be tricky in certain corner cases. For example, there are some subtle differences as far as logging P1 is concerned: In the FAA world there are situations where no less than three persons can log P1 simultaneously. No such thing under JAR-FCL/EASA, where only one person can be (and log) PIC.
If you logged P1 under one of the situation where the FAA allows you to log P1, then those P1 hours might not be used as PIC hours for the purpose of revalidating an EASA SEP rating, for instance.
And the ultimate corner case is of course if you use that FAR 61.3 provision, and fly an N-reg within the UK on a UK-issued license. Do you log according to the FAR or EASA ruleset?