Multiple misfires temporarily on cold start

[vonxolo]

2009 528i E60

My car has been doing this rough idle, misfires, for a couple minutes in cold starts.
During this issue the gas pedal is nearly unresponsive. After a couple minutes it just starts to run normal.
I’ve recently replaced the fuel pump, and am pretty sure I need to replaced the filter and sending unit.
Could this be related to the filter? It seems more electrical by its behavior, like the fuel module possibly?
And help appreciated

[Schomosport]

After a host of troubles, where cold start roughness was the least of my troubles, I replaced the HPFP with a new one. Since replacement I get a little rough running on a cold start, less than half a minute, where the engine runs rough and if you try to drive off straight away it bogs down and won't pull the skin off a rice pudding. I find giving a quick rev immediately after starting helps. After the first half minute it is no trouble the rest of the day. I have aduff nox sensor, too tight to change it since I don't really understand what casued it to fail and they are expensive. I suspect it might be injectors but alreadyb spent so much money that so long as the fuel consumption remains OK and it passes UK MOT I live with it. (2007 330i). Had similar with a 335i, that tended to misbehave a mile down the road but soon cleared. BLack art these things.......