@SanGamo - that's exciting!
Do you mean Skinners (Boys Grammar School) or St John's Primary School? From your list I would avoid Temple Grove, I know a couple of people at Southborough and Pembury and they have been happy there but if you are living in Tunbridge Wells they would not be convenient to travel to - bad traffic in that direction.
Broadwater Down has historically not had a great reputation but it has now been taken over by the leadership team from Langton Green who are amazing and the area around is being regenerated. It also has a nice outside space and is fairly easy to get to if you're living in TW, especially the south side. The school intake there is more diverse than at other schools. From your list above this would be my recommendation.
I would avoid The Wells, having moved a child from there as the outside space is not good and the leadership / general organisation is quite lacking IMO. Claremont you are very very unlikely to get a place unless you live literally next to the school and even then you might wait years on the waiting list.
I have heard that St Marks is a nice school, also in the south of town and easy to get to with good outside space. It is more religious than most though if this is something you wish to avoid. It really does depend where you are looking to live. I wouldn't waste your money on the Mead though - their reputation has really gone downhill since it was bought by venture capitalists and their facilities are worse than most state schools in the area.
I always hear good things about St John's and St James schools, also St Peters. But their catchments are quite tight to you would need to live in the area and get on the waitlist. Happy to answer any more questions about schools and areas to live - it's a minefield!!
Good luck :)