Andrew, just ignoring for one moment the dual nationality matter, if your wife has already been in the UK not illegally during the past three years (which appears to be the case) then she is entitled to apply for naturalisation
now, given that she is married to a British Citizen, you! So this dual nationality matter is a real live question for you now ... not one you need to defer until some later date.
That is, whilst most applicants for naturalisation need to have had their ILR for a minimum of one year, for the spouse of a British Citizen the test is rather different ... they merely need to have ILR ... the length of time they have had ILR does not matter.
So Korean nationality law! Obviously a question of thinking that through. Not an easy choice. As you say having a British Passport also gives Citizenship of the EU and lots of rights to live and work etc in other EU countries.
I shall just make this comment. If your wife indeed entered the UK on her student visa at least three years ago, if she applies for naturalisation this month, October, she will avoid the need to pass a Citizenship Test, which applies to applications received by IND on or after 1st November.
I presume your wife's English is up to standard. For example, did she need to get an IELTS score before coming to study in the UK? If so that IELTS certificate will undoubtedly be sufficient for the purpose ..... the required English standard for naturalisation applicants is ESOL Entry 3 level .... which is equivalent to an IELTS score of merely 4.0!
Now a serious decision to make!
