Re withdrawing the judgement, I am not aware of any mechanism by which this can happen. If it was, the case would have to be heard again from scratch with a new panel. This is also what would happen if the EAT finds that the ET was biased and sent it back. There is no way Kemp, Brown and Russell (the tribunal member, not the barrister) will have any involvement in a re-hearing.
Some of the tribunal's findings of fact are based on their incorrect interpretation of the law. If we accept the SC judgement, Upton should not have been in the female changing room and therefore SP was not proselytizing when telling him this, she was stating the law. Those should fall away.
However, unless the EAT feels that the tribunal's findings of fact are so out of line with the evidence as to be perverse, the findings in relation to the credibility of the various witnesses, etc., will stand unless the case is sent back to the ET.
By the way, to return to the Bryson point, it was agreed that SP had referred to the women's prison incident. The tribunal decided that was a reference to Bryson. Whilst that doesn't feel like an unreasonable finding, it occurs to me that a more neutral finding would be that it is not clear whether SP intended to refer to Bryson, but it was reasonable for Upton to think that she was.
This judgement is so poor that it clearly will not stand. Those rushing to use it to say that men are allowed in the women's toilets and changing rooms need to pause. Unless the SC think their judgement does not mean what it seems to say, at some point the lower courts will be told clearly that they must stop trying to subvert FWS and that they should stop any advocate attempting to relitigate that decision.
I don't know for sure whether SP will ultimately win on all counts, but I hope she does. She may not think of herself as a campaigner, but she is one now.