When you undo the screws, the controller hinges around the 2 lugs on
the top and can be removed, leaving the back-plate and connections.

It is just the Honeywell box that wobbles but I suppose I will find
out why soon enough.

That's what I meant. The controller has two screws to allow it to be
fixed to a single flush-mounted backing box. You've got a double
surface mounted box, so only the l.h. one of those two screws is
holding the controller onto the backing box.

Not sure what a tundish is.

A funnel. It directs the discharge pipes from the 2 safety valves into
one outlet. It allows you to see whether either valve is dripping. Also
(this is the important bit) if the external pipe is obstructed (by
physical damage, freezing, vandalism, etc) it allows the safety valves
to discharge if a fault occurs. The discharge will go over the floor,
but if there's no discharge then the cylinder could burst explosively.

There is a large copper pipe (22mm?) that runs down the outside wall
to
ground level which I assume is the safety overflow.

Yes. There should be a separate discharge pipe for the boiler safety
valve, with it's own tundish. You need to ensure that 22mm is adequate
for the unvented hot water storage system's safety valves, when the
tundish has been fitted (see the Installation & maintenance manual).
Make sure the immersion heater has the proper thermostat with a
secondary manual-reset thermostat. Or get someone competent & certified
to do it for you. You need to get those things properly serviced
annually.