However hot the day, jumping into the river in central London is not currently recommended! (Even the moral rural stretches of the Thames can be risky...) It's a tempting idea, though, as the river becomes ever-cleaner - if only a safe swimming space were available.
Indeed, it's not a new idea. The Thames did have a lido in the past, nearly got another one for the millennium, and could have one again if architects Studio Octopi are successful. They have launched a project to build swimming baths near Blackfriars Bridge - and yes, the water will be warmed and filtered. If the thought appeals to you, then you can support the project on Kickstarter and receive rewards involving chances to swim in the new pool if and when it opens.