Update in case useful to any fellow DIY ignoramuses..
matt1e was right - the stat on my top element was broken. When the water heats up but doesn't stop heating up till the safety thermal cutout on the stat pops, that's a good clue where the fault lies.
If your bottom element works OK, an easy thing to try is to put the stat from the bottom element into the top element. The stats are dead easy to change - you don't need to drain the cylinder because the stat rod fits into a watertight sleeve.
***Make absolutely sure all your power is off!*** then disconnect the elec wires from the connector screws, and gently pull the stat directly out, horizontally towards you. It's only held in by a pair of clips. You might have to waggle it a bit to loosen it.
This is what it looks like when pulled out.
This shows the thermal cutout - tiny red button that pops out when the element overheats. This is your first thing to try if you have problems - press it in again with a pen point, it should give an audible click if it has popped out. But be careful and keep an eye on your cylinder if you turn it on again afterwards - the cutout is a safety feature to deal with overheating, and if the heater's overheating there may be a problem with your stat that must be attended to.
The temperature setting in this pic is where you set the general heat you're aiming for - mine was set to about 4, which was fairly hot. The power connectors are where the elec wires connect the stat to your mains.
This is the back of the stat showing the two prongs that hold it in and pass the mains current to the element.
From what I've been reading on the interwebs, these Cotherm stats on Heatrae Sadia Megaflos are prone to fail after a while. Apparently they've brought out a new version to replace them - it's white instead of black. If you buy a stat it's probably worthwhile making sure you get one of the new type.
If you need an element, you can get element + stat for about £45 on eBay.
Side note -- when I cancelled my second booking with Pimlico Plumbers they wanted to charge me for the parts they'd ordered from their supplier. But I found there are some excellent rules called "Cancellation of Contracts made in a Consumer's Home or Place of Work etc Regulations 2008" which mean you can cancel a contract made in your home within 7 days. So you're not obliged to pay suppliers for such orders, and you're entitled to a refund if you paid a deposit. See
http://www.which.co.uk/consumer-rights/buying-services/how-to-cancel-a-contract/your-rights/ for details.