In recipes online I see recommendations for buns such as Martin's Potato Rolls. However it doesn't seem like it's possible to get this kind of bun in the UK, or indeed anything other than the basic no frills sesame seed burger bun. What is the best burger bun available? And where should one shop for it?
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After living in the UK for more than a decade I share your pain in finding good burger buns. You can often find burger buns in the store but they are usually pretty bland and artificial. Better quality buns in the right shape or size are usually heavy and chewy.
You could take a cue from the artisan "posh" burger places that are popping up all over London and use brioche buns. These are widely available, and they have many of the right qualities as they are soft but still have structure and flavor. They do have a sweetness which I don't think goes with a burger but that's really minor. Other than that it depends on what supermarket chains are around (Waitrose and coop are the best sources I've found) and what bakeries you have around you. Shop around and pinch some buns till you find what you want.
You could go down the route of making them yourself, the Hawksmoor slider bun recipe is available on-line here, it's pretty simple and reliable.