Depending on where in Crystal Palace you might have a fair wealth of charging options- just east of the palace itself there are a load of lamppost chargers (Ubitricity and Char.gy branded) both of which are able to be used with the Shell recharge network.
There are also a few rapid chargers around there which should give you a decent top-up and get you to a decent state of charge.
Your 15 mile commutes shouldn’t be taxing but you’ll need to charge similarly to how you would charge at a petrol station most probably- depending on which car you might get away with weekly, unless you can use the on-street lamppost charging.
Christchurch looks a bit more limited in terms of charging options but there are still a few- a couple of Rapid chargers (think a half hour/40minutes to go from close to empty to close to full (10/20%-80% or so), there are also free, slower chargers in the big Tesco down there by the look.
You should be fine to make the trip from one to the other in one hit as long as you have a decent charge, but speed on the motorways will ear into range (faster is much less efficient) particularly depending on the car, and the weather (the colder the weather the lower the range).
Which car are you looking at getting?