Can I fly an N-reg helicopter in the UK with an LAPL(H)?
That will depend whether the FAA will accept the LAPL as a 'current pilot licence' in accordance with §61.3 as the licence does not comply with ICAO Annex 1. It will certainly not be accepted for issue of a licence under §61.75.
With this LAPL(H) do you have to do it on all the types you are rated on?
There are no type ratings on an LAPL(H); instead, the validity of the licence is based on a minimum recent experience:
FCL.140.H LAPL(H) — Recency requirements
(a) Holders of an LAPL(H) shall only exercise the privileges of their licence on a specific type when they have completed on helicopters of that type in the last 12 months:
(1) at least 6 hours of flight time as PIC, including 6 take-offs, approaches and landings; and
(2) refresher training of at least 1 hour total flight time with an instructor.
(b) Holders of an LAPL(H) who do not comply with the requirements in (a) shall:
(1) pass a proficiency check with an examiner on the specific type before they resume the exercise of the privileges of their licence; or
(2) perform the additional flight time or take-offs and landings, flying dual or solo under the supervision of an instructor, in order to fulfil the requirements in (a).
Note the requirement is 6 hours PIC and not 5 as suggested above.