The current system is probably configured so that the room thermostat only operates the pump marked CH that circulates the hot water around the radiators. So unless the thermal store is also hot, the pump will only circulate cold water around the radiators. [The other pump marked HW is a bit of a misnomer, it is responsible for heating the Heat Store which is required for both CH & HW, so 'HS' (Heat Store) would be more appropriate.]
One of the features of Hive / Nest is that they are controllable remotely, so if you wanted to use it to turn the heating on as you were heading back home after a few days away for example, you wouldn't get any heat unless the heat store was on and hot, but from what you say yours doesn't have a timer and is on 24/7 anyway, so shouldn't be a problem for you.
The NestE is a good suggestion if you don't want to install a mains supply, but it is a 'cut down' version of the standard Nest, and doesn't have all of the functions of the standard Nest, as a result, some have been disappointed with it. And it is just a single channel to replace a standard room thermostat.
One of the features of Hive / Nest is that they are controllable remotely, so if you wanted to use it to turn the heating on as you were heading back home after a few days away for example, you wouldn't get any heat unless the heat store was on and hot, but from what you say yours doesn't have a timer and is on 24/7 anyway, so shouldn't be a problem for you.
The NestE is a good suggestion if you don't want to install a mains supply, but it is a 'cut down' version of the standard Nest, and doesn't have all of the functions of the standard Nest, as a result, some have been disappointed with it. And it is just a single channel to replace a standard room thermostat.