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My Vaillant system claims to do that, but it doesn't have dynamic input of pricing so you have to type it in on the head unit. I'm not sure it's even possible to get an API for dynamic pricing of UK energy prices.

Octopus have a tariff with dynamic electricity pricing: https://octopus.energy/smart/agile/

There's a (third-party?) dashboard of the data here: https://agile.octopushome.net/dashboard

46.73p per unit at peak time while the average is roughly equal to the regular tariff? That seems like a bad deal.

Does make it very clear where the afternoon peak is though, and that's a nice dashboard.

There are hybrid heat pumps that do exactly this.

Bet you could do this in home assistant

Yeah, if I could figure out the calculations to use and source all the data...