Wow, I don't envy you - getting across London will be tough enough with the cat carriers nevermind the train journey to Newcastle. When I lived in the UK I used to take that train quite often (I lived in london but I'm from Newcastle originally) and I can't imagine doing it with cats (both in terms of space and meowing cats, just too stressful!) If flying isn't an option, is there no way you can drive? I know it's hard after a long trip but it might be worth looking into.
Sorry if I sound negative.

Maybe if you travel at a quiet time (tues, wed, thurs are best) and if you book extra seats for the cats it will be fine (try and get one of the tables with 4 seats so you can all be together). Good luck!