It's a bit like the answer to where to find good beer in LA ("get on a plane to Portland") - London doesn't have a great cask culture for various reasons but like NYC it's big enough to have good - if expensive - outposts if you know where to look. But what it does have is some of the great pubs - see this for a starting point :
https://pubheritage.camra.org.uk/pu...=5&perpage=100&nistatus=&orderby=distance:asc
And see some past threads on the subject :
https://www.homebrewtalk.com/thread...-london-help-appreciated.672199/#post-8720093https://www.homebrewtalk.com/threads/uk-trip.664719/#post-8573539https://www.homebrewtalk.com/thread...r-favorite-recipe.472464/page-45#post-8509714
So much depends on the detail of when and where you will be - good cask beer relies on turnover, so a place with great beer on Friday night may serve vinegar on Tuesday morning, whereas the Harp is a good option for those slower sessions as it's always kinda busy but just gets insane on a Friday/Saturday night. A lot of the Square Mile is shut at weekends, the Bermondsey Beer Mile historically was only open on Saturdays but most of them are now open most of the week except Monday/Tuesday, and so on. The
London Brewers' Alliance festival is at Fuller's in Chiswick on 17 June, although I think it tends to be more keg-y than cask-y.
In general avoid Greene King pubs - they're fairly ubiquitous but the beer's meh and cellarmanship is...variable. Shepherd Neame are OK, Fuller's pubs are generally quite good but there's only so much Pride I can drink, Sam Smith's have some of the great pubs - the Princess Louise, Cittie of York and Cheshire Cheese are obvious examples - and whilst they are the only mainstream brewer to stick to wooden casks, they only do one cask beer and it's OK, but no more. Then you have breweries with random outposts in London, like Harvey's Royal Oak near Borough Market, and the Bricklayers by Putney Bridge which is sort of a Tim Taylor tap in the capital.
So it depends on how much time and how much drinking you want to do, and how constrained you are by other members of the party - for instance if they wanted to go to Fortnums or the Royal Academy then I'd say go to the Red Lion behind Fortnums which is a Fuller's pub with a CAMRA 3* interior but I wouldn't necessarily go there as my one pub in London.
But you can't really go wrong on a weekday by starting with Pride and Harvey's Best at the Harp, then either going via the Lamb & Flag (to drink) or Cheshire Cheese (to look) to the Princess Louise and Cittie of Yorke on High Holborn (have your one pint of Sam Smith's OBB at the Louise, then just have a quick look inside the Yorke), then drink at the Olde Mitre off Hatton Garden, go via either St Paul's or drink at the Blackfriar (by Blackfriars Bridge), stop at the Pelt Trader under Cannon St station if you want something more modern, pick up a bottle or two of Fuller's to take home from the Waitrose at the north end of London Bridge, then cross the river to look at the George on Borough High St, then drink Harvey's at the Royal Oak. Then if still thirsty, head on to Bermondsey to experience one of the hearts of modern British brewing, with the likes of the Kernel, Cloudwater, Brew by Numbers and Anspach & Hobday (their London Black is becoming the go-to alternative to Guinness).
But if you are eg taking the train from Edinburgh, then there's some good options around King's Cross, if you're taking Crossrail/Heathrow Express in to stay around Paddington there's options there, it all depends.