explain how British heating systems work in the first place

Not to take this off onto a tangent ('cause that never happens), but....

It sounds a lot like most US hot water systems, other than two things. One, the hot water heater isn't usually inside the house. It's almost always in a basement or crawlspace. Although I used to live in an old apartment that had one about the size (linear cabinetspace-wise, anyway) of a dishwasher. Two, the way you put it, it sounds like it's on a timer, which the ones in the US are not. They just detect the temperature of the water in the tank and heat if it's too cold. I'm not an expert, but I think it's probably about half-and-half gas units versus electrical units, with the electrical ones being immersion heaters.

There's a new trend towards on-demand heaters, but I've never seen one outside This Old House, which is the US's premier DIY show, not that anything they do any more could be done by yourself, assuming you even had the mansions that they do it to.