A complex area, which can only be given definitive guidance via the overseas desk at London AFCO.
In all cases single nationality Commonwealth citizens can only serve in specified RN/RM trades, much the same as single nationality UK citizens can only serve in certain restricted Commonwealth military service trades, in some cases single British nationality citizens cannot serve in any military trade. Generally Dual British/Other nationalities can serve in a wider range of trades.
The reasons behind this are to do with security protocols & in the case of the British submarine world , the US/UK agreement with regard sharing & access to nuclear classified material. Oddly, dual British/US citizens must give-up their US nationality to serve in the UK submarine service - how bizarre is that?
Whereas residency waivers to join are certainly a consideration in the odd case, Nationality waivers are non-negotiable unfortunately. The only grey area that occasionally crops up is for Irish Nationals who have already served several years, but even then, you need a witch doctors ju-ju bag & chicken bones to work out comprehensible/understandable rules which may or may not apply.