It course it would. Unlike Spain the U.K. population is happy with the principle of self determinism (Chagos islands are somewhat special as is an American base with no population for the last 50 years, historical claims to land are somewhat different)
Realpolitik sometimes gets in the way of this - why doesn’t the US recognise Taiwan as a country? Why doesn’t Brazil recognise Kosovo?
If Scottish people want to be independent, or northern Irish, or Cornish, or Yorkshire, or Falklands, or Gibraltar, or Ceuta, or Catalonia, or Crimea, or Texas, or Kosovo. or North Cyprus, then that’s upto the people who live there.
Northern Ireland had a referendum and one side decided as they wouldn’t win they wouldn’t take part. Scotland had a referendum too and decided to stay. There’s been no real change in opinion in Scotland, despite the travesty of the EU vote.
The problem with NI, Scotland and the EU is the opinions is roughly 50/50, whatever you do half the country are losses off.
In reality if the people of Northern Ireland wanted independence they would only want it if they could join Ireland, and joining Ireland would require the Irish government to accept a basket case of an economy, not to mention the continued partisan issues that have plagued the country for centuries, nobody wants change, but as predicted in Trek, an act of terror (brexit) may well lead to it as brexit changed the status quo.
Catalonia wanted complete independence, although as you said though, less than 40% of the adult population wanted it in that vote. Certainly they should be allowed if the majority want it, as long as Spanish citizenship is not removed from any resident who doesn’t want it removed and every resident got Catalonia citizenship as well if they wanted it (my view would be you would only grant independence if over half the adult population agreed to forfeit their Spanish citizenship).
Catalonia would then be free to apply for EU candidate status as a free and independent country, just like Scotland. Spain would not want that, so Catalonia would have to be a small independent country and start from scratch.
Spain doesn’t recognise Kosovo either. Their view is that territory is more important than self determination. See their view on Gibraltar. Spain would thus be unlikely to recognise an independent Scotland or Northern Ireland, and thus EU membership as an independent country would be out.
So with Ireland not wanting the economic drain of the north (only 1 in 8 Irish Republic citizens want unification if United Ireland had to pay), Spain not wanting any independence move to succeed, Northern Ireland is stuck in the continued compromise it has been since 1997. Brexit of course massively damaged everything, and far more want the U.K. to join the EU than want say Catalonian independence, but we are where we are.