Looking at the timeframes, the process would have to work very quickly for any Russian or Belarus athletes to be at the world cup series.
The definitive registration deadline for Cairo, the first, is 15th December and nominative registration 15th January. Cottbus is next and it’s 22nd December then 22nd January for them. I don’t know how it would work in practice with late definitive registration anyway, assuming Russia and Belarus can’t enter anyone while the ban is still in place. But even on the assumption it’s allowed, the approval process would have to work promptly. I know gymnasts qualified for 2022 worlds on the basis of winning one world cup, but Olympic qualification is surely going to be much more competitive.