From CAP804
The UK National Private Pilots Licence (NPPL) is a licence issued by the CAA that is valid in UK airspace for the piloting of UK registered aircraft only. (The NPPL may only be used in another country with the permission of the relevant authorities of that country.)
I think that a reasonable person could interpret that as,
is valid in UK airspace for the piloting of UK registered aircraft
only.
Also as
is valid in UK airspace for the piloting of,
UK registered aircraft only.
Further down in CAP804 the NPPL is defined as
A non-ICAO licence
which permits the holder to fly for leisure within the UK only, within the terms of the SSEA, SLMG, or Microlight rating as applicable
So, if I am flying an N-reg, my licence is not valid. The FAA (allegedly) say that they permit me to fly their aircraft, if I have a licence (issued by another country) that would allow me to fly that plane. However my CAA issued licence is clearly not valid for N-Reg, so I don't in fact have such a licence.
The FAA might well allow me to fly it, (as in, they don't object) the CAA do not say that the licence is valid for non UK registered aircraft. Do the FAA have authority to permit me (NPPL only) to fly a 172 in the UK?
The CAA document seems clear enough, " the piloting of UK registered aircraft only " so I don't see how you can use it fly an N-Reg C172 here.
So, if we do fly it, are we doing so under the FAA rules rather than the CAA rules?
CAP804 is not worded as "valid only in UK airspace" but "valid in UK airspace.. for UK reg... only" which is a different circumstance.
Like so many other parts of legislation relating the NPPL and LAPL, this seems to have been drawn up by blind nuns. How hard can it be to write the rules to reflect the intent? The exact wording is important, as is getting it right.
My interpretation is that the CAA do not explicitly allow it. They don't prohibit it either, they just don't include the flying of N-Reg aircraft as a privilege of holding an NPPL.