Hi Charmania,
You sound like a very good Mum who wants the best for her kids!
However you will find that just because someone went to Oxford or Cambridge doesn't mean they will be a good English teacher. I have three good friends who went to Oxford - one is a physics graduate, one is an IT graduate and one is a music teaching graduate.
None of them would be as good at teaching your children English as many others I know who did English degrees or teaching degrees at lesser-known Australian universities!!
Feel free to make this your criteria, but I would suggest that a better place to start would be to look for someone with a degree teaching English as a second language.
All the best!
HKAussie