Hi!
The 'Kingston' area is generally considered a 'nice' area to be in. It's very hard to give feedback based on a written description, but I'd say that if the house is clean, well maintained, maybe recently refurbished or has decent high quality fixtures and fittings etc, is in a quietish street with similar familes to yours living there then £1550/month for a 3 bedroom place seems like a very reasonable amount.
Like any place in the UK, areas and their 'tones' can change within a very short distance - just a few hundred yards or a couple of streets in some cases!
I used to pop over to Kingston when friends were at the university there, and so they were in the area in student housing, compared to the areas I lived in at university (Bradford, W.Yorks) these places were fantastic! and the areas were nice too, the quality of Kingston and surrounding areas carries on with family type residences and of course the high end as well. So long as your place isn't too close to the student areas even if there's a few nearby, you shouldn't find them too much bother.
The general area itself I think is very nice - Twickenham, Teddington, Walton on Thames, Esher etc are all nearby and very nice as well. As you get to know the place, Worcester Park is a very desirable area just on the other side of Kingston.
Overall, I think you've found yourself a very reasonably priced, good home to rent from your description, and I think you mentioned £2K a month allowance previously, so use the rest of the dosh to pay for the council tax and utilities if you can!
Good luck with everything!
Cheers! DtM! West London & Slough UK!