@Datun
Are appeals usually upheld by leapfrogging over a High Court refusal like that?
Yes, what will happen is that they will make an application to the Court of Appeal and must show why the decision in the lower court was wrong in principle.
The only time you can appeal is where the decision of the lower court was wrong in law or on the facts; or it was seriously unjust because of a serious procedural or other irregularity in the proceedings in the lower court or the result would represent a substantial injustice.
Their application will be looked at by a single Lord or Lady Justice of the Court of Appeal who will decide if they have an "arguable" case.
If they have an "arguable" case they will be granted permission to appeal, if they don't then they will be refused permission.
I think that @Abitofalark made a very important point:-
It's an indication of how seriously the High Court viewed this case that it sat with three judges rather than one and even more when you look at who they were: one was the President of the Queen's Bench division, the most senior High Court judge in that division and one was a Lord Justice - a judge of the Court of Appeal - and the third a High Court judge.
When you look at who was sitting in this case and the detail of the judgment and also the detail of the reasons for refusing an appeal (the latter linked to by the OP in this thread) I think it unlikely that any Court of Appeal judge would find an application for appeal would be "arguable". But, who knows?
I think that the fact that the President of the Queens Bench was sitting is very important (she is actually also a Lady Justice of the Court of Appeal as well so also sits in the Court of Appeal) as well. So, effectively, there were two judges from the Court of Appeal who were deciding this case (albeit that they were sitting as judges of the High Court at the time).
I really cannot imagine the Court of Appeal even thinking about overturning this decision.
What little experience I have is in the area of family law and, in a similar manner, this would be like a decision of Sir James Munby (the former President of the Family Division and also judge of the Court of Appeal) being overturned by the Court of Appeal - it really just never happened.