No, charter schools are not the same thing as grammar schools. Grammar schools are selective schools that take a certain percentage of students based off a test. Otherwise, grammar schools are run exactly like the other schools (called comprehensive schools) with the same funding, etc. There are also academies, which are run a bit differently -- those are closer to charter schools, but still much much more run like regular schools than charter schools in the US and most don't have tests. And remember that no matter what kind of school your kid goes to, everyone in the UK takes exactly the same tests at the end (GCSE, A-Levels) which are set by the government.
I think you're getting thrown off by mumsnet. Just as in the US, most kids in the UK just go to their ordinary (comprehensive) school, and like in the US, some are better than others. Depending on where you live, the comprehensive schools may not be as good because the grammar schools might skim the "best" students from the top, but even that's not always true. If you find a place you like, that is not in an area with a lot of grammar schools, and read the Ofsted reports of the schools near where you want to live (reports are flawed, but still pretty useful), you'll likely find a school that works. If not, just as in the US, find another place to live closer to a school that does.
IMO schools are less uneven in the UK than in the US because the funding is not based on property taxes. Moving to a wealthier area may mean a different kind of of parent group, maybe more money from the PTA, etc. but it doesn't mean the school itself has more money.
So the UK definitely has schools that "both non-selective and supportive of an education beyond 11 years of age." Many (most?) of them are just this! But just like in the US, you'll have to do your research and plan where you live wisely.