As @PeterGreen says: If the mother was a British citizen born in the UK, then her children would be British citizens. You can be British citizen without having a British passport. So if the children are British citizens then there is no problem whatsoever. (Or the father might be British, which wasn't mentioned).
If the children are EU citizens, then they should be able to move to Britain without problems as well.
If the children are not British citizens and not EU citizens, and the mother is in the EU, then I believe that EU freedom of movement + Surinder Singh ruling allow the mother to move to the UK and bring her family. EU rules are important because British law alone wouldn't allow her to bring her non-British family. If living outside the EU, she could move without problems to any EU country for some time, and then move to Britain.
I don't think having the second parent in the UK is either required or helpful in this situation.
Some good information if the mother is british citizen by descent, and the children not British nor EU citizens, is available here: Which type of UK visa a child of a British citizen by descent need to apply for in order to settle in the UK?