I was thinking more about the recent Midterms, YMB, where Democrats took control of the House of Representatives (the effective equivalent of the House of Commons) by a hefty margin. As it happens, the result in 2016 is also vaguely relevant -- the simple fact is that the US President has only ever appealed to a minority -- but it's possible to ignore that and *still* claim, quite correctly, that the Democrats have a mandate to deny the President funding for his wall.
Neither side can really afford to back down. Well, more accurately, Trump *could* have backed down earlier, and probably nobody would have really noticed (least of all his core support). But now he can't. The catch about trying to be a strongman is that *any* sign of weakness destroys that image. Better to be a sensible leader, and accept that there is no support for the wall but still a mandate for other of his policies.
Oh, and in any case, unless people forgot, the Mexicans were going to pay for the wall anyway. So why does he need the US to pay for it?