A single hose even coiled or zig-zagged isn't going to be particularly efficient. The best solar panels have around 80% efficiency meaning that they absorb and transfer 80% of the sun's energy into heating the water that flows through them. These solar panels have many thin tubes in parallel and close together. Some even have inside curve channels to create circular flow in the tubes for greater heating (though this only improves efficiency by about 5%). In the U.S., solar panels are independently rated by the
Florida Solar Energy Center. The most efficient panels are the FAFCO Revolution at 1063 BTU/sq.ft. though most panels are in the 1000+ BTU/sq.ft. range which is reasonably efficient (these BTU/sq.ft. are heating amounts in Florida -- you won't get anywhere near this amount of heating where you live).
The main problem you will have is that many of your days are cloudy and the air temperature is not that warm. This not only has the sun's energy hitting the panels be lower, but also makes the panel's less efficient since they will lose some heat if the water temperature is warmer than the air temperature. As shown in the chart in
this post, on a cloudy day a water/air temperature difference of just 5ºC lowers panel efficiency from 80% to 46% on an already lower heating ability.
Finally, there is wind. These black flat mat panels don't do well if there is a water/air temperature difference and there is any wind. The efficiency drops rapidly in that case.
You can use
this link to find out the amount of solar energy you get in your area. In southern England, for example, the peak solar is in June and July at around 4.8 kWh/m
2/day for a horizontal surface. If you tilt your panels towards the sun (about 28 degrees), then you can increase this to 5.4 kWh/m
2/day. Your 15' x 42" pool at around 618 cubic feet or about 4625 gallons (about 38,550 pounds of water) would need 38,550 BTU (11.3 kWh) to raise the water temperature by 1ºF. So assuming a generous 80% panel efficiency, you would need about 2 square meters of panel area for every 1ºF of temperature rise in one day. A typical 4'x12' panel is around 4.4 square meters.
Your pool surface area itself is 177 square feet (16.4 square meters) so using a clear bubble-type solar cover is the least expensive approach to heating your pool and would be needed anyway even if you were to use solar panels so that you could retain heat at night. If your pool vinyl liner is dark, then you might get reasonably high efficiency of sunlight heating the pool, though probably not more than 80% efficiency at best between reflection from the vinyl color and the absorption in the solar cover. So just the solar cover alone could result in a temperature rise of up to 8ºF per day, at least initially (i.e. when the water is the same as average air temperature) though in practice you might see only half that. A good clear bubble-type insulating solar cover should be able to increase your water temperature to at least 15ºF higher than average day/night air temperature if not more. In June/July, the average day temperature in London is around 22.5ºC (72.5ºC) while at night it is 14ºC (57.2ºF) for a rough average with more day weighting of 66ºF so just a solar cover alone could get your pool water to 81ºF (27.2ºC) or more.
Of course, if you live in the northern part of England, the achievable temps will be quite a bit lower.
