If the servicer changes, the old one will inform you in writing where to send payments and who to contact with questions. If you already sent your payment to the old lender, no worries. You won’t incur a late fee because there’s a 60-day grace period following the transfer. The new one should also send you a notice with the same information.
So all you have to do is prove that you made the payments to your old lender. The new refinance lender will submit your proof to a "rapid re-scoring" service and your credit will be cleared.
However, it looks like there is more than this to the story. If you made the payments but they were not processed, you should still have the money and so there would be no reason to "catch up." Just send the money for the missed payments, and they should remove any late payments that were the result of their error.
If your mortgage payments did not transfer for weeks, and you did not contact the servicer, and then spent the money instead of paying the lender, you have a different problem. Because now, you actually ARE late with your mortgage. I would get the money together to completely catch up and ask the lender to agree in writing to remove the late payments once you repay the past-due amounts, since they have some responsibility here. They don't have to, but they might.
