Hey OP,
Glasgow born and bred and I love my city. PP have raised excellent points - scenery/ walking/mountains/ lochs/ culture/ vibe/ diverse/ multi-cultural/ friendly/ community feel/ smaller than London but so much to do/ great public transport/ parks/ teen activities and independence.
I'm South Side (still v much the city, though) and teens catch a bus 2 mins from house or train 5 mins and are in town within 15/20 mins. DS1 17 goes to as many gigs as humanly possible! They can easily cycle too. They cycle to rugby club/ mountain bike in local park.
Posters suggesting East Dumbartonshire/ East Renfrewshire are not listening to you wanting some 'grit'. As nice as these areas are it wouldn't be me living in Bearsden/Milngavie/ Newton Mearns/ Whitecraigs etc they're too suburban and very, very middle class for my taste: I'd suffocate. It's very much keeping up with the Joneses. You'll get that type everywhere, I suppose, but there's definitely greater numbers in those areas.
If I had your budget I'd buy a bloody lovely house in the West End- Dowanhill/Kelvinside/ Kelvinbridge Partick/ Thornwood/ Broomhill/ Jordanhill. Nothing beats it for me. My sister lives there, and although I love my area, if I had the funds I'd move in a minute. Yes, we call some of them the West End Wankers, but there's mostly normal folk.
Schools are plentiful. I'm not a subscriber to this 'You must go to Jordanhill/ St Ninian's' etc, it's nonsense. Good schools are not the exclusivity of the upper middle classes. My DC1 got 5X A band 1s at Higher, the school dux and due to get the same results at Advanced Higher in an ordinary, local, very mixed comprehensive. Scotland's comprehensive system works. You have dedicated staff in every school.
West End Secondaries include Hillhead/ Hyndland/ Notredame as well as the aspirational Jordanhill.
I adore Northumberland - Tynemouth/ Cullercoats/ Bamburgh particularly, stunning and super friendly - but don't know Newcastle.
Don't know Leeds well. It does seem near lots of things, but a good colleague ran a mile from it. She's from Yorkshire but trained in Glasgow. After 8 years here she got a job in Leeds. She lasted a year and 'Ran as fast as I could back to Glasgow'. She said it just didn't compare: was too small and has no culture to speak of! Only her view, she does love theatre and art, but worth considering.
PM anytime if you need advice about areas.