I am a little confused about your marital situation. I believe that you are implying that the boys’ father is a Taiwan citizen, correct? Is his Household Registration in Taiwan still current, or has it expired? Where is he now? Is the marriage still valid, or has there been a divorce?
For the time being, I will assume that the Taiwan father is working in the UK, and you are still married. If the marriage is not registered in Taiwan, and there is no mention of the “Taiwanese father” on any existant paperwork (filed in conjunction with the children’s residency applications), then it is certain that they will be regarded as pure foreigners, since they hold UK passports and have ARCs stating that they are citizens of the UK. Hence, they could only apply for PR at the age of 20.
On the other hand, if the existence of the Taiwan father is known, and the marriage is registered in Taiwan, then the picture changes considerably. Under the current MOI interpretation, the children could not apply for PR at all. This becomes a serious problem at the age of twenty, when the children can no longer renew their ARCs based on a “dependency to a parent” status.