Have immediate family members in Singapore. Can my family of three immigrate there?

Your sister is a British citizen and has lived in Singapore for more than five years. I believe she is also a permanent resident of Singapore. Only the immediate family members of permanent residents, such as parents, can apply for a family permit, and the residence period can be up to five years, depending on the validity of the passport.

In your case, neither you nor your children can apply for family permission and stay in Singapore through their relationship with your sister.