Another place to look if you want a lot for your money and a lovely area is Low Fell in Gateshead. Demographic and community is similar to Gosforth but you get double the house for your money. It’s not on a metro line but I know several Low Fell families with kids at various private schools that get picked up by the school bus laid on by the school. Particularly Dame Allen’s.
Whickham is also worth looking at as that’s a lovely semi rural area with good transport links
I echo what was said above about Jesmond the areas around the private Schools are generally quite nice and not too student orientated. I personally love the young demographic in Jesmond but it’s not for everyone!
If you want a good state school to have a look at - Gosforth High is excellent as is Emmanuel College on the Gateshead side of the river. Emmanuel has an entrance exam but it is not selective. There is also a superb state high school at the coast.
Wherever you choose the area is extremely well serviced by public transport so your kids will have no issue getting to whichever school you choose for them. If you are in catchment for a good state school and opt for that, then all the schools have a fleet of scholars buses that are specifically for kids at that school.
I would definitely shop around for private schools as they are all very different and what suits one of your kids might not suit the other. I agree with the summations made above about the private schools in the area.
RGS/NHFG very academically selective
Dame Allen’s a bit academically selective and separate girls/boys school
NSFB and Westfield not selective
Durham School not particularly selective
I have friends with kids at all these schools and they are all very happy with them.