The children need a "family of a settled person" visa. Even though their mother is not yet settled in the UK, she qualifies as a settled person because she is a British citizen.
The category to select in the image is "Join partner or family for a long stay."
SztupY has given useful information about the requirements and the high cost of this visa, as well as about another option, which is for the children to enter on a standard visitor visa and then apply for leave to remain. This is also an expensive option. For two children, you're looking at fees in the range of £2,000 to £3,000.
If the cost is prohibitive, or even just a significant factor, there is yet another option, however, commonly known as the Surinder Singh route, which involves moving to another EU or EEA country for a period of time. When you do that, you can move back to the UK under the EU freedom of movement regime instead of the UK immigration laws.
If you qualify for this, you will be able to move your children with you to the UK using an EEA Family Permit, which is free of charge, and then apply for a residence card, which costs £65 per person.
To qualify for the Surinder Singh route, however, you need to be able to show that you have established yourself in another country under the EU freedom of movement rules. Obviously, for some, the intangible costs of moving to another EU country for several months or more may outweigh the £2,000 or £3,000 extra that they would spend to move directly to the UK. For others, the prospect of living in another EU country for a year or two might be appealing. In other cases, the intangible costs might be negligible, for example if they already live in another EU country.