Secora secundaflora (Texas Mountain Laurel)
I planted one of these lovelies at my last house. They are marvelous landscape shrubs/trees.
https://www.youtube.com/watch?v=nnVwgEsdzZs. I love purple blooms so much. They smell like bubble gum to some or like grape KoolAid drink to me. Mine grew 15' before I sold that home and moved to a new house.

The 2 planted here by previous owner were young shrubs 4 years ago when we purchased the place. I've shaped them up more like trees so they won't overtake the walkway (which they were about to do, as they sprawl if left to their own devices). I just topped them 3 weeks ago to get them to fill out some bare spots and I don't want them to get too big and shade what little sun I have out front of the house. Below is a south-facing photo of them (flanking the sidewalk). Full shade in mornings; full sun midday; hot, afternoon sun the rest of the day.
I don't know if these evergreen plants are even available in the UK, but if they are, they are well worth a garden consideration. They tolerate oppressive heat, drought, hard freezes, even the occasional snow. They seem to require no fertilizing to bloom nicely or much care at all (other than pruning if you want to shape them away from their natural sprawling shrub growth tendency). True survivors, these.
Sorry I don't have a photo of them in bloom to share (happens early spring). A recent computer crash cause me to lose that photo.

My low-carb recipe site: https://buttoni.wordpress.com/
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Which got me thinking about our joint ‘hot, dry shade under trees’ conundrum, what about a hardy plumbago as a ground cover plant? Info here:
https://www.highcountrygardens.com/perennial-plants/unique-plants/ceratostigma-plumbaginoides
I was interested to hear (on another thread) you thought Portulaca would grow in shade, must try it...
There is also a gorgeous deep pink dianthus growing wild in the shade of trees near me, something else to explore...
Link below says it is borderline hardy here.
Probably winter wet rather than cold temps does for it.
https://www.shootgardening.co.uk/plant/sophora-secundiflora