I don't think this lot will take over (and, with all his faults, I doubt that Gove is really a true believer in this sort of thing; the others are more worrying).
But we should never be complacent. In 2008, I was tutting smugly about the American Republicans choosing Sarah Palin as VP candidate, and was sure that the religious right would never have influence here. In 2010, I was reeling as the MP for my seemingly-liberal, later strongly Remain-voting, constituency, was smeared by Christian Right activists as 'Dr Death' for being too pro-choice, and defeated probably as a result: certainly the fundies took the credit! We managed to reverse this, and now have a possibly even more progressive MP, who was almost the only non-Tory MP in England to increase her majority in 2019. But it's always necessary to be vigilant and not to just assume that things 'can't happen here', even in quite progressive places. And of course many parts of Northern Ireland in particular are influenced by the religious right at any time.