Have watched The English on Amazon Prime in UHD and it is also oversaturated same as on iPlayer UHD.
Recorded The English on my Aura in HD and colours are not as saturated. Colour space is BT 2020.
If other iPlayer UHD streams display as you would expect then it must have been a choice of the
production company to have filmed The English in this way.
Colour space on 1080i/p HD is / must be Rec BT 709. Programmes will be ingested played out and encoded suitably.
UHD ingest of the same programme for iPlayer will be the wider
BT 2100 and HLG is used for iPlayer. (2100 is based on 2020- in its most recent versions incorporating the HDR function definitions from a quick scan read of stuff).
See
https://www.bbc.co.uk/delivery/technical-requirements
UHD delivered programmes may be mastered in other HDR formats (within BT2100 spec) but must be delivered to BBC as HLG (and BBC supply lookup tables for the conversion from PQ HDR formats).
I suspect, but can't be sure that BBC will require both HD and UHD files are delivered from the programme makers (so they are responsible for getting it right 'artistically') and only SD stuff gets downscaled for emission and conversion from Rec BT 709 to Rec BT 601 (aka CCIR 601) for PAL/NTSC/Secam (if necessary, the last two having extremely similar colour spaces) during code and mux.
Modern TVs have so many 'per input' settings and adjustments that are only invoked when watching a suitable input source/stream that I'd not entirely rule out a 'difference' in such TV adjustments entirely for those who've observed this?.
{Nor that BBC haven't mis-labelled something in metadata nor had a recoding error for certain devices' streams only. Or even that the programme maker converted to HLG incorrectly before delivery... but I'd have expected BBC to notice that sort of error.}
I can't find
The English on Prime TV... but haven't tried too hard.