I've just put my flat on the market, having lived here (SW London) for 10+ years. After seeing a few of my friends move during the stamp duty holiday, I somehow got the itch to do so, too -- albeit too late, it seems...
I've found that there's not that much supply where I'm looking (Surrey villages), though my own street has a lot of supply (a dozen places or so). I have seen a number of properties reduced in price as they weren't shifting, but equally have seen some lovely ones move to SSTC within a day. It's just weird!
I've resigned myself that if it doesn't happen and I stay put for now, it's fine. It'd be nice, but not a necessity. Besides, it wouldn't be that much more space than I have. I only realised recently that my (small!) one-bed condo in the US has the same -- or more! -- square footage than most of the two-beds I'm looking at here!