The FAA says that "if you have a licence to fly the aeroplane in your country's airspace, then you can fly an N reg in your country's airspace without holding an FAA PPL"...or words to that effect. It does not mention anything about ICAO compliant or anything that I can see.
Unless the LAPL specifically prohibits flying anything other than a G registered aeroplane, then as far as it goes, you are legal....Just as pointed out earlier, the IMCr can be used in an N registered aeroplane in the UK, the IMCr being sub-ICAO.