As the title: what is the most appropriate projection to reduce distortion, if I'm just showing the UK on a map?
Is WGS84 fine (since it's a relatively small area), or should I use some other kind of projection?
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If you're publishing this for viewers in the UK, the “right” CRS is EPSG 27700 (or 7405, if you're working in 3D), as Vesanto has already said. The OSGB grid does distort the islands, making it hard to tell that Edinburgh is actually further west than Bristol. But UK folks are used to this, on the mainland, at least.
There isn't a single UTM zone that neatly spans the UK from Belleek (8° 5′ W, Zone 29) to Lowestoft (1° 45′ E, Zone 31). Using unprojected WGS 84 kinda smears the islands out a bit. Your choice very much depends on your scale, your audience and whether distance or area need to be kept accurate.