Updating iCloud ID and password

Hi

I recently updated my Apple ID and password (I used the same information when I set up my iCloud account). I did this on my iPad. However, I am repeatedly asked whether using my iMac or iPhone, to enter my iCloud password for my OLD ID. I'e tried to find a way of making the same changes on both these other devices but I am failing miserably. Anyone able to help?

Many thanks

Alan