Your children's father is a Brit and you are married to him - you can apply for Brit passports for them because they are entitled to British citizenship. Then you needn't worry about visas for them, only for yourself.
As you've been married for 9 years, you should be entitled to Indefinite Leave to Enter, rather than the 2-year visa. That means no more paperwork unless you want to then apply for citizenship yourself after you've been in the UK for 3 years.