Hot water boost

OK, this is for the geeks who really know their stuff.  How does the hot water boost buttons work?  Do they heat the water line directly?  Or do they do a shortened fill-and-heat cycle of the combi boiler?  We've been here 6 months and we still haven't figured this out!

I am confused - I have never had a hot water boost on a combi boiler 

I did have a Hot Water boost when I had a old heating system in which I used to have a hot water tank, but when they removed the hot water tank and gave me a combi boiler instead, the button in the cupboard became redundant.

A combi should heat water directly, without the need for a cylinder.  If you have a cylinder (a torpedo-looking thing) in a cupboard or utility room, then you have either a regular or system boiler.

OK, I went searching throughout the flat, but I didn't find any cylinder!  The boiler is identified as an Albion Maxistore direct combination boiler (link).  Is this not a combi boiler?  The landlord told us that it was a newer combi boiler, and we never questioned that (didn't even know the differences!).  The boost button is a Horstmann ElectriSaver E30 (link).

The boiler you linked to isn't a combi - it's a storage tank.  These work by heating water at night when power is cheaper and then using it during the day.  To be honest they are a bit rubbish and I thought they'd died out years ago - I'm surprised how common they are on this board.

The control you linked to is a timed control that switches its self off after a while.  I'd guess it's connected to the heating element in your boiler - kind of like the element in a kettle - and basically turns the heat on for 30 mins to boost the temp of the stored water.

OK, I went searching throughout the flat, but I didn't find any cylinder!  The boiler is identified as an Albion Maxistore direct combination boiler (link).  Is this not a combi boiler?

If this is what you've got, then that's not a combination boiler.  It's a hot water heater which stores heated water on the bottom and has a cold water store on the top.  It's all starting to make sense now as you said in another thread that you have storage heaters.

If this is the one you have then the heater heats the hot water during off peak times (overnight).  If you run out of hot water after bathing in the morning, then you'd need to hit the boost button to re-heat/re-fill the hot water portion of the tank.  This could take up an hour to re-heat and is re-heating on peak electric tarriff.

Does this make any sense at all?

That does make sense.  I was asking this mainly because it seems that the shortest time frame (30 min) doesn't seem to do anything to the water!  My wife and I keep debating as to whether it heats the water line directly or if it refills/reheats the tank itself (either way, we don't get any more hot water).

Depending upon the model of Maxistore you have from the link quoted there are several options for the way the system works overall.

You might have the model which is just the hot-water cylinder and which is fed from a separate cold header tank in the attic, or you might have the model which has an integral cold header tank sitting directly on top of the cylinder (common in apartments, flat-roofed homes etc.).  This is where there might be some confusion with terminology, since these can also be called "combi" units, but are not the same thing as the newer combi-boiler:

http://www.albionwaterheaters.com/CombiUnits.html

Whether you have a separate or integral header tank won't affect the other options though.

If you look on the Albion page, you'll see that the Maxistore cylinders have provision for two electric immersion heater elements.   With a hot-water storage cylinder, because the hot water tends to naturally rise to the top as it warms, they are always arranged so that the cold water enters at the bottom and the hot water is drawn from the top.   

A common arrangement is for the lower element to be controlled through a timeswitch on the Economy 7 power, so that it switches on overnight during the cheap rate and gives you a full cylinder of hot water by morning.   The upper element is then wired to either a regular switch or to a manually started timer such as your Horstmann unit, so that you can get a boost during the day should you use all the hot water.  Because it's operating the element in the top half of the cylinder, it provides a smaller quantity of hot water in a much shorter time, and thus more economically than running the bottom element and waiting for whole cylinder to warm up (useful if you've run off all the hot water for a bath and then just need some to do the dishes later).

Sometimes though you'll find that instead of two elements only a single one has been fitted at the bottom and the boost timer then operates on that one as well, which would account for the longer warm-up times.  Have a look on the side of your cylinder and see if you can find one or two elements (usually the visible part on the outside will just be a round-ish cover, often black but sometimes other colors, into which an electrical cord runs).

Those cylinders which are listed as indirect models also contain a coiled pipe inside which is connected to two separate bosses on the outside of the cylinder.  These are intended for connection to a central heating boiler, so that the hot-water from the boiler is pumped through the coil to heat your water (there is no mixing of water; it's purely heat transfer from the hot coil).   

What do you have for your main heating?  As you say you can't find a boiler anywhere, I'm guessing you're probably all electric for both room and water heating.

Edit: 
Nevermind that last question - Just seen on your Consumption thread that you're all electric with storage heaters.