PIP Moving from Northern Ireland to England

I moved from Northern Ireland to England last month. I gave the NI office my new addesss in England the day before I was due to move. I recieved my PIP payment a few days after I moved but it still had the old social security agency info on my bank satement. I thought it would have dwp or something similar on the payment?
Is there anything else I need to do if the NI office have been informed of my new address? will they transfer my case to dwp in the background? Im due to my next payment soon so dont know if itll have sorted itself out by then in the background.

Hi
I phoned the NI office first and they confirmed they did have my new address that I gave them a month ago. But, that I would have to phone DWP myself so that they could pull it through. Is this information correct? I thought the NI office would do this electronically or will I phone to phone Dwp myself to confirm I've moved?

Any tips on how I handle DWP? I didn't know and nor was it explained that I had to phone them directly myself otherwise I would have a month ago.

ok thanks is there a recommended time to phone when they are less busy? I phone twice at after 5 and was on hold for 10 mins and then 20 mins.