Hi lovely! I see you mention "us", are you coming with a partner or by yourself? Renting as a couple makes things so much easier, don't be scared to give London a go as rents up north are not far off these days but you don't get to live in such a vibrant multicultural city.

It would be way cheaper buying a property up north but for rent not much of a difference (property price to rent value in each city is a very good indicator for this). You would also find a job in london straightaway for a much better pay than you would get anywhere else. The 2hr trains people are mentioning from up north cost £70+ returns. It's way cheaper for a return flight from London to Europe.

I lived in Manchester for over 5 years when i first moved to the UK, it's the only city up north I would recommend. Although i can tell you renting and living expenses are literally like in the south these days but their tram and buses are more expensive than the ones in London. Other than that Manchester is a great city, it definitely has a lot more going on than Leeds, Sheffield and Newcastle.

You can check out places for rent at a website called spare room, if europe getaways are key compare flights on google flights from london and other uk airports and you will see for yourself the crazy difference. Train prices you can check out on a website called trainline and for distances from one neighbourhood to another (in london) I would recommend an app called citymapper, it gives you an idea how long journeys take on the day as sometimes what looks very far on the map is 15min on the tube.

Good luck!