I think it was o series where Anton Dohrn was claimed to be uks highest mountain at 1700m aprx. There is a larger mountain in uk antartica, mt Hope which is currently confirmed at 3239m ans haa been confirmed as highest. I actually called this out while watching show and was suprised when Sandi named Anton Dohrn. In2017 it was confirmed in meters and was shown to be 377m higher than previously thought. This would still make it higher than Anton Dohrn at time of filming
You are quite right to say that Mount Hope in Antarctica is higher than any other mountain in what is claimed as British territory. But two issues arise:
1. Do we consider territories as part of the country? While (for instance) the Turks and Caicos Islands are a British Overseas Territory, we don't usually think of them as an integral part of the United Kingdom.
France thinks differently here, and does consider its overseas possessions to be integral parts of the country. But that is not the standard thinking in the UK.
2. All territorial claims in Antarctica are subject to the Antarctic Treaty. That treaty does not strike the claims out, but it does prevent them being enforced in any meaningful way.