I am with Lloyds bank and while I was changing all my documentation with various bodies, they accepted income and outgoing payments with my new name AND old name while things were being 'tidied up'. If you explain to your bank, they will understand as all they want is your money in the long run. Honestly, my bank were great about the whole thing and made things easy. They left this open for a while I believe. Within a couple of months all my stuff was changed everywhere and that was that. The bank will not push sensitive information in the opposite direction, if you see what I mean, so you should be safe with HR. There will come a time though when this will have to stop.
Hope that helps and good luck.
Nikki