My personal take is that a child who performs at a certain level in the English SAT, would be able to score similarly in History or Geography.
Same with the Sciences...the Maths SAT would roughly correlate to the child's Science ability.
At least, that's been the case among my DD and her friends, which I realise is a small sample.
In terms of independent schools/those that don't do SATS, I would get a vibe from the school, look at work, talk to the teachers...that's how I would tell whether it was a reasonable fit for my kids. However, what I am personally looking for for my children is a high degree of academic performance. A lack of external data to judge this would put me off a school.
That being said, DD is now 16 and her state comprehensive school is so focused on everyone getting the best academic results that she does miss out in other ways. So there are no trips, no plays, no dance shows, no music recitals etc. There is some sport, but not a wide range. However, she does a lot of extra curricular which fills this gap. This is why I say that for MY kids, it's a great school. Others may not see it that way.