I have lived - and taught - in counties with no grammars whatever, and also in a county with a small number of 'residual' grammars, attended by a tiny proportion of children overall.
Proportion of children at private schools is probably slightly lower than it is somewhere like Kent (mainly due to demographics - at least a couple of the counties I am talking about are less well-off than Kent on average, so private schools are out of the reach of more families IYSWIM).
Yes, there is some migration to areas with the better comprehensives, but less than you might think, especially as most of the counties are relatively rural with distances being large, so moves to better catchments turn into HUGE commutes and the differences between the schools are not sufficiently large to merit the effort IYSWIM. With a very 'polarising' environment such as that in Kent, everything becomes much higher stakes. In the counties I am better aware of, the differences between the schools are not that vast so the pressure to move for a better one is reduced.
Oh, and minor point. Setting by subject, not streaming. Streaming is pretty daft - how do you stream a child who excels at Maths but struiggles in subjects that require writing? Much better to set for separate subjects, so such children are in appropriate classes for every lesson.