If you can work out how to get to either Stoke or Uttoxeter which I think are the closest mainline stations you can get the train with 1 change to Edinburgh or there are some direct trains from Derby. www.nationalrail.co.uk
It takes around 5 hours but you won't do it any quicker than that!
Just make sure you've got plenty of things to do to keep the little ones entertained on the journey!
Don't know what time you are travelling but there are buses to Uttoxeter & Hanley both of which have train stations. The former would probably require a change of train at Derby & the latter at Crewe. Overall travel time won't be much different at around 6 hours so go with whatever gives the best deal
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