That's just tranny scare-mongering. There's nothing in the judgment that can affect access to birth control.
Contraceptive drugs have been extensively tested to see 1) if they really work as contraceptives and 2) don't have any weird side-effects. All of that is extensively documented in respectable journals and thoroughly peer reviewed. There's a public paper trail.
Puberty blockers on the other hand are a random collection of drugs, licensed for other uses, that some very dodgy doctors have been handing out to kids without telling the full story. Not only has no research been done, but they actively discourage it, sabotaging, doxxing and forcing out anyone who even thinks the R word.
Download the judgement and note how many times the judges were "surprised" that the Tavistock did fuck all to follow up what happened after they drugged the kids. They were carrying out an experimental procedure but couldn't be arsed to do all that science-y stuff like collecting and collating data. Give kid pills, pose for photo, get to go on pride marches and meet Emma Watson. All that statistics stuff is so, like, twentieth century.
There were two lines to the judgment. 1) These kids can't really have a clue what they're giving up. 2) Even if they could understand that, the doctors haven't got a clue what they're selling.