Some good ideas here, thank you.
I like the starting point that the judgment doesn't change the law, all it has done is clarify it, and now we know where we stand we can think about where to go from here.
From a legal point of view (IAAL), I would make the point that the Supreme Court is the highest court in the UK and therefore the judgment cannot be appealed. The law is the law, and the only way to change the situation now is for parliament to amend the Equality Act. (Subtext: it is what it is and we all have to accept it unless and until new legislation is passed to change it.)
From a political point of view, I would say that the Supreme Court judges are neutral and politically independent (this principle is woven into the fabric of the judicial appointments system in the UK, unlike in the US where Supreme Court judges are politically appointed), and their only role here was to interpret the legislation. The legislation itself was introduced, debated and enacted by the last labour government. (Subtext: anyone who thinks this result is due to shady right wing forces interfering in British democracy is talking out of their arse.)
^ This point could be particularly useful if the subject comes up when talking to Americans, because they may not understand that our judiciary is completely independent of our political system and made up of experienced judges who interpret our laws impartially, not like in the US where Donald Trump gets to stuff the Supreme Court full of his batshit crazy friends. We actually have a grown up legal system and so what the Supreme Court says carries weight and can be trusted.
In terms of the issues discussed in the judgment itself, the one I like the best is where it explains that if you create a single sex space or service for women on the basis that you have a legitimate need for it, allowing some biologically male people to use it (who have male bodies and are likely to be perceived as male by many of the female users of that space) undermines the very rationale for its existence as a single sex space.
Say women are A, men are B, trans women are C, and so on. Each of these groups have their own different needs.
What C wants may very well be a shared space with A. But if that isn't what A wants, the shared space is only working for C and not A. And if it isn't working for A, you have lost your legal justification for excluding B.
The judgment makes it clear that all groups matter, and that A and C's needs should both be accommodated, but C's "needs" cannot include being in a shared space with A if that is not what A needs. Otherwise C's needs are being prioritised over A's.