tl;dr: heat pumps are widely used for swimming pools, can probably be used for hot tubs, and should work reasonably well in your climate.
Swimming pool with cover
I couldn't find good information on hot tubs, but I did find that the U.S. Department of Energy has this page on heat pump water heaters for swimming pools. Then, a quick search suggests that Seattle is the city with the climate most similar to London, which is in the south of England.
For a pool with a cover, a heat pump (HP) pool heater will use 1700 to 3300 kWh per year. This assumes that you're using it for the months of June to August, and that the HP has an average COP of 5.0.
Depending on what the HP replaces, this usage will represent savings of 40 to 80%.
Hot tub vs water heater
Three things change for your situation with a hot tub:
- Year-round operation
- Hotter temperature
- Less water
In this case, it starts to look more like a storage water heater, so I started looking at some of the data that impacts COP of heat pump water heaters (HPWH).
[+] Advantages of a hot tub vs HPWH
Bigger is better. According to an NRDC study, HPWHs with a larger tank have a higher annual COP, especially in cold climates -- in North Dakota (site of the coldest average installation in the study), an 80 gallon (300 L) tank was about 15% more efficient overall than a 50 gal (190 L) tank. A hot tub is obviously bigger than a water heater.
Lower setpoints are better. See chart below from a NIST study. For a hot tub, the "tank" doesn't need to be as hot as for a water heater.
[-] Disadvantages of a hot tub vs HPWH
- Cold climates are worse. From the NRDC study cited above, The HPWHs studied were in basements, whereas yours will likely be outdoors. However, they found that the colder the basement, the worse the COP -- see chart below. So for a hot tub that's outdoors, you'll probably get a worse COP than for a water heater.
- Warmer inlet temperature is worse. A water heater is heating up cold tap water, whereas a hot tub will be re-heating the water already in the tub, which means the COP will be reduced -- see chart below from a study by the Universidad Politécnica de Madrid. However, you don't need to add as much heat since the water is already closer to the setpoint temperature, so the total impact on energy usage may be minimal.
Since heat pump water heaters are already marketed for swimming pools, it appears they would already be available and reasonable for a hot tub as well. I couldn't find any research data on heat pump hot tub heaters, but there's a lot of literature on heat pump water heaters for domestic use. A hot tub heater will have some advantages and disadvantages compared to these, but it's probably safe to assume that the COP will never drop below 2.0, and could reach as high as 5.0 in the summertime.