I grew up calling my friends parents and neighbours Uncle and Aunty, but it has died out in British culture (my son didn't do it).
In other cultures it's alive and well, and as your children are part of both cultures, I think they should learn the right etiquette 'rules' for each culture. MIL shouldn't insist they call British adults uncle or aunty. But you shouldn't stop them doing what is polite in their Dad's culture, either.
They can learn two sets of rules, easy.
I was in a relationship with someone from a culture like this and his kids did this. With their British family and friends first name terms. With anyone from their Dad's culture, Uncle and Aunty. It is seen as being very disrespectful otherwise, and it's their culture as much as the British side. In a way it was quite nice - I rather liked being called 'Aunty' myself by all the others younger than me (makes a nice change from getting the mick taken out of me as I get from my own neices and nephews!
) I also got called 'Madam' and 'Maaa' a lot, which was peculiar at first as it would sound like a piss take from someone British, but its a way of showing respect.
I understand that they are growing up in the UK so the main cultural influence is British. It doesn't cause any harm, and it's a positive thing as I see it, for them to also learn about the other culture.
If you were to live in your partner's country, I'm sure you would want your children to learn British manners alongside the ones prevalent in his country, so they are well equipped in either environment.