Some countries (eg the USA) have a very explicit requirement that citizens enter and leave the country only using their US passport, even if they also hold another passport.
Other countries, including the UK, AFAIK do not.
However, that is not the point.
The point is that you should get your children their British passports asap because that is, de facto, the one and only sure way to prove their British citizenship.
This thing really winds me up, but most Brits do not realise the sheer lunacy of rising up in arms against ID cards (which almost all of the rest of the civilised world has) while passively accepting a perverse system in which in theory you don't need compulsory ID, but, in practice, how on Earth do you prove you are a citizen if you don't have a passport? Windrush has shown that you don't, and that your government may even deport you, even if you are a citizen, because, well, without a passport how do you prove it???
I know of families who have had to dig up work contracts and pay slips of 20 years ago, to prove that the parents were legally settled here when their children were born, and so the children were born British. If they hadn't found those old records, the children would not have obtained a British passport.
Oh, one more thing: the Home Office does issue citizenship certificates, but they cost more than a passport!!!!
A passport is even more indispensable after May's hostile environment policies, which were the main cause of the Windrush scandal, and which are likely to become even more 'hostile' following Brexit .