I've been looking into the "using the single user toilets will out us" argument a bit more. The GLP's appeal application refers to the Supreme Court case R (C) v SSWP [2017] 1 WLR 4127, a trans person taking on the DWP for the way frontline staff could determine their trans status, and how that was unlawful. They lost. That case itself says at 45:
For the first time in this court, and somewhat faintly, the appellant argues that the DWP’s policies, and specifically the implicit “outing” involved in the SCR, create a “harassing environment” contrary to section 26 of the Equality Act 2010. ... This is not an allegation which can sensibly be made in a claim for judicial review of the DWP’s policies in relation to transgender people. ... it is quite clear from the DWP’s efforts to understand and to meet the appellant’s concerns within the bounds of practicality that its policies aim to have the reverse effect: to respect the dignity of transgender customers and to avoid creating an intimidating, hostile, degrading, humiliating or offensive environment.
I don't think it's much of a stretch to extend this argument to toilets: providing unisex toilets is done to respect the dignity of transgender [people] and to avoid creating an intimidating hostile degrading or offensive environment.
I want to stick my neck out and say that firstly the C of A will concede that GLP did have standing (just as SM should have been seen as having standing in the JR of the Hamptead Ponds policy) but that both the HC and CofA will both deny permission to appeal the case on the grounds of there being no realistic possibility of success.