How soon do you start the new job and do you need the two pay cheques to run concurrently?
S/he’s not going to get two paycheques concurrently unless someone feels very generous!
OP is resigning, not being fired. OP needs to either work out their notice (in which case they’ll be paid by old employer and not by new one because they haven’t done any work for them yet) or the OP’s old employer will waive the notice period but won’t pay them (because OP didn’t work!).
Edit: I think I might have misinterpreted your post and I take it back. I suppose it is possible that e.g. you had an old employer that paid 4 weeks in arrears and a new employer that paid 2 weeks in arrears that you could in fact get paid twice during the same week. But you wouldn’t get paid twice FOR the same week unless you actually worked two jobs during the same week. Apologies for confusion.