I don't have direct experience - my dd starts at Teddington this year - but I have a lot of friends with children there. I don't know anyone at Turing House but I did visit it so can share my impressions.
I think both are good schools, and in other parts of London less spoiled for choice parents would be fighting over them. I thought Turing House was very impressive when I visited: obviously, being new and so still very small, it has a real family atmosphere and the children were really impressive. I really liked most of the teachers I met, though was bemused by the science teacher saying that girls are naturally drawn to biology...
If it was staying in Teddington it would be hugely oversubscribed, I'm sure. But many parents - including me - didn't want to choose it when there is still so much uncertainty about where it will be located. I felt very uncomfortable with the local politics, and it ending up somewhere it's basically not wanted and the potential for local resentment about admission policies. Plus, because its catchment is similar to Waldegrave, it is predominantly boys and likely to remain so. But if it gets certainty over location this year, I would certainly recommend you visit it - I hugely liked a lot about it, particularly the caring intimate atmosphere.
Teddington is our local school, so started points ahead. I like its building, I like the atmosphere, I like the fact that all the children I know who are there seem happy and feel safe. I think it probably skates along on its demographic, and doesn't add as much value as it should. I worry about the staff turnover and I wonder if children would be allowed to float along under the radar. It gets talked down quite a lot by local parents - but funnily enough, I've only heard this from parents whose children aren't there. The parents whose kids are there seem pretty happy.