@confusedlots , do you have any experience of home educating? Most parents don't find the pre-GCSE stage a problem.
The GCSE curriculum is more or less self-contained, so it isn't the end of the world if you don't cover exactly the same content as school in the run-up to it. Of course, if there are areas your kids are really struggling with, which you don't want to learn alongside them and which will affect other subjects, you might consider a tutor in person or online for a few hours a week.
OP, if you are looking for a short-term solution then you absolutely can just remove your kids immediately in order to keep them safe and happy. The next few months are not critical. You can keep them ticking over in any subjects you feel they need to keep up with, or take a break and branch out and learn about something entirely different, following their particular interests about something they weren't able to learn at school. If you go with what excites them, learning isn't "work" and you don't have to battle them over it.
Do you work full-time? You're right that being stuck at home every day could be dull for your kids, but still possibly less bad than the bullying they were experiencing at school. If they are independent enough to get out on their own, they can go off swimming, skateboarding, to the library, walking the dog, visiting neighbours or friends etc. They can continue with any clubs or sports they already do in the afternoon and evening and weekends, or take up new ones. How do they usually spend their time during the school holidays; can they do the same sort of things now?
Longer term, next autumn is make-or-break time for your eldest. Nowhere is the inflexibility of the school system more apparent than during the "GCSE years". Schools can't comfortably accommodate new arrivals much after the start of Y10 and really don't know what to do with kids who turn up on their doorstep later than that. So you need to get your older child into a school in the autumn or make home education work.
If there isn't a viable school available, there are various options for home educating Y10 and Y11. For example, many families just do core GCSEs. Five or six exams are typically enough to get onto the next stage of education or employment. You can prepare for them with an online school or a tutor-led home education study group or independently. Some colleges have a 14-16 programme where your children might be able to pick up some GCSEs. Some kids postpone GCSEs and do them at college after 16, though the college GCSE choices are more limited than you'd get from school or home education.
If I were you I would take them both out immediately and take some time to help them recover from their difficult experiences at school. Don't stress about the academics for the rest of this year. Home education is much more efficient than school, and catching up is easier outside of a classroom. By law they do need to be learning from day one, but that doesn't have to be formal education. Then you can see what the prospects are of getting your older child into a suitable school by the autumn, and/or explore alternatives for carrying on with home ed instead.