I am sure such schools are out there. I think DD goes to one (though south of M4)
(DD is in an infants and Junior school system down here with approx 60 in each year).
Yes, there is a friendship bench and some of the yr2 kids are made "play supervisors" each term. They seem to tai it very seriously (my shy daughter seems a bit like the Pied Piper these days so goodness knows what she did) The school also pops a few toys (balls/hoops/skipping ropes into the playground and the lunchtime supervisors encourage the children.
The yr2 kids moving up to Juniors have exchanged letters and met their buddies. EVERYBODY is in the end of year play and XMAS play. EVERY child is expected to participate in assemblies (one class a week does an assembly..each child has to speak some lines out loud , but it can be in a group). I'm not sure that this sort of thing appears on the school website, though. I'm not even sure it pops up in Ofsteds. Maybe ask on your Mumsnet local (I believe other sites for local Mums might exist too).
I'm not sure whether you think your DD isn't interested in forming friendships with her classmates (I suppose they might all be horrible, or all in cliques) or that she just can't figure out how to.
There are a couple of children in DD's Yr2 class who genuinely seem happy to play by themselves (in their own imaginary worlds, quite often). I know one mum is slightly freaked out that her youngest DS is like this but the rest of the class (certainly my DD's group of friends) seem to accept this and still like him...don't query it. They happily include him when he fancies joining in for a while.
Are you a working Mum, or do you manage to do school pick-up/drop-off once in a while? The school holidays are about to start so I think I would ANNOUNCE to DD that she can't spend the holiday without at least one or 2 classmates coming round for a playdate. I feel fairly confident that I have the email and mobile/landline numbers of al DD's class. If you don't have that sort of system, then get it set up. Even if only 5 or 6 families agree, it is a start.
If you are worried that playdates might be a big "fail" maybe offer to take another kid to the cinema (Odeon and Vue do cheap kids films in the holidays)....if you are in Salford there must be a cinema quite close.
Or swimming/summer school clubs etc. They aren't all super-expensive.
I freelance, but I do like offering to take a few kids to one activity if another parent is prepared to ferry to another.
We are in a fairly small town here, but i like the fact that my DD has friends outside school (swimming/ballet/gym etc, but I guess that all costs).