There's a lot to unpack here and I'm not sure I agree with their conclusion.
In the end it all comes down to what you mean by a beach, what you mean by closest and what you mean by London.
Some would argue there are beaches inside London so there's no need to look beyond its borders.
Ruislip Lido's beach, for example, has a perfect patch of sand ideal for towels and toddling. But the water it backs onto is a reservoir, the sand has been carted in from elsewhere and the whole thing's probably too artificial to be counted, so let's move on.
What of beaches along the Thames instead? This photo shows the waterfront at Barnes with tiny locals ambling around below the shoreline, but I could have shown you any number of places where riverbed access is possible as the tide falls. One of the most central is the foreshore outside the Royal Festival Hall which definitely has a bit of sand in it so should that count? The waterfront below the Tower of London opened as Tower Foreshore in 1934 for the benefit of scruffy East Enders seeking beach-type activities, but that closed in 1971 so let's cross that off. Folly House Beach on the eastern side of the Isle of Dogs is a strong contender, and will have seen many 'beachgoers' during this recent hot spell, although that's a converted dock so probably doesn't tick the right boxes. But if everything below Teddington Lock is tidal so technically the coast, arguably London has dozens of Thamesside beaches.
As you head downstream, or rather downestuary, the claim that these are beaches gets stronger. This photo shows Aveley Bay - an actual bay for heaven's sake - amid the marshes between Rainham and Purfleet. There are admittedly signs onshore saying Danger Soft Mud so a game of beach cricket would be ill-advised, but the river's edge is certainly rubbly enough to support a picnic blanket. What's more the Greater London boundary actually divides this bay in half so the closest beach to London could be infinitesimally close, in Thurrock.
If I consult my Ordnance Survey Explorer map of the estuary all of this so-called 'beach' is coloured beigey-grey, meaning mud, and the first thin patch of yellow for 'sand and shingle' doesn't appear until you reach Gravesend. I can't confirm this in real life, alas, because it's been high tide every time I've visited. Also Gravesend's official tourist portal makes no mention of a beach, golden or otherwise, so I suspect this may be cartographically far-fetched. It's another 10 miles before any more yellow intrudes.
Ah yes, All-Hallows-On-Sea, the wannabe seaside resort on the Isle of Grain. This is a proper beach at last and closer to the capital than Southend, which is why the Southern Railway built a railway spur here in 1932 hoping to swipe some of the daytripper trade. But All Hallows never caught on, then WW2 snuffed out all further dreams of expansion, and today the footprint of the station is a caravan park jealously protecting the golden sands from anyone attempting to drive in. At 33 miles from Charing Cross it could perhaps be the closest sandy beach to London... were it not for a contender on the opposite shore that's only 30 miles distant.
Welcome to Canvey Island, specifically Thorney Bay, a horseshoe indent in the sea wall close to the Leisure Island Fun Park. I can confirm it has a sandy rim because I've seen it, a long steepish yellow bank where sandcastles are perfectly possible, and a texture underfoot that the burghers of Brighton and Hove can only dream of. This is the beach that Which mentioned but dismissed, if you remember, and I can see why they did because it's hardly the ideal target for a great day out. But that's not the question they originally asked, they just wanted the closest beach to London, which suggests Thorney Bay ought to be the true answer.
The closest Blue Flag beach to London is on the Southend Riviera, specifically Westcliff-on-Sea, which is described in the Blue Flag citation as "a gently sloping stretch of golden sand and shingle." This has the added bonus of being just beyond the official end of the Thames Estuary, the Crow Stone, adding authenticity as a beach genuinely lapped by waves from the sea. Shoebury's beach is better and Southend's is busier, but the Leigh/Chalkwell/Westcliff end (50 minutes from Fenchurch Street, or just 30 from Upminster) has the benefit of being closest to London by rail.
Of course Which weren't happy with Southend, of course they weren't.
But Frinton's an hour and a half away by rail, for heaven's sake, and more than 60 miles if you choose to drive. It's definitely not the closest beach to London which I'd say is Thorney Bay on Canvey Island, but in the end it all comes down to a matter of definition.