Don't stress unduly - King's Cross to Paddington is an easy 20 minutes. And, even with separate tickets, you are covered in case of disruption (your tickets will be from Edinburgh to London Terminals and London Terminals to Cardiff). If you find you miss your connection, you can board a later train - but do go to the ticket office first to get your ticket endorsed - and you may have to insist on this (I can quote the precise regulation if you need it...)

Let me know if you need step-by-step directions to get you between the two stations - it's a journey I do a lot.

Are you on the train that came from Aberdeen? Ouch, looks like there's been some nasty problems up there today - train fault followed by flooding...