Our current house (3 bed detached, double glazing, reasonable insulation) has a gravity fed heating system powered by an old oil fired boiler. It’s reliable (touch wood) but clearly not the most efficient. For example you can’t have the heating on without hot water.

Hot water is stored in a copper cylinder in the airing cupboard, the tank currently has no thermostat fitted and has an electric immersion backup.

We have a wireless thermostat which we can move around the house as we see fit to optimize when it comes on and TRVs on most rads.

The HW cylinder appears to have a bulge in it which clearly isn’t good. This will need replacing soon.

Are there any improvements we can make to increase the efficiency of this system while we replace the tank?

(This likely isn’t our forever home so i’m reluctant to rip out the old (>10yrs) old potterton boiler and replace it with a new combi style one.)

Fit the best insulated HW cylinder that you can and insulate as many pipes as you can reach. 45C minimum is needed to keep legionella at bay so consider how this will be achieved, especially if you run a shower from the cylinder

Consider whether you want to add independent HW and heating controls, zone valves, etc. If you go down this route then the immersion heater is arguably less important

Have a look and see if the cisterns in the loft need a clean out, covers fitted, etc

You only need to hit 45 degrees occasionally, but its a pretty low temperature anyway so shouldn’t be an issue.

You say your insulation is reasonable…that probably means you haven’t got the full 270mm in the loft. Heavier curtains plus one in front of any external door onto a living area.

Always shut the curtains in the evening, we have a large double glaszed bay window in our front room and the difference between opened and closed is significant…on the rare occasion it gets uncomfortably hot we just open the curtains halfway and it cools quickly.

HSE recommend storing water at 60 degrees and water at outlet at 50 degrees within 1 minute to prevent Legionella.

Most importantly too, prevention of warming of cold water systems above 20 degrees is required.

Are there any improvements we can make to increase the efficiency of this system while we replace the tank?

nicely insulated tank.

fit a zone valve so that the HW can be run on a timer (even if it is still a gravity loop) and switched off when the heating runs.

anything else on the plumbing side will quickly start getting expensive.

personally, I wouldn’t touch anything else if its all working, the reliability is worth quite a bit in my book.

Get rid of the copper cylinder and fit an unvented cylinder (you can get a used Megaflo on ebay for £200 if the new price is too much) – this will transform your hot water supply.
Keep the old boiler if it’s still working OK, paying for a new boiler/fitting that will be dead in about 10 years is utterly pointless.
Have valves fitted so you can use the boiler to heat just the water in the summer – unless you have PV in which case fit a proportional controller to heat your water using spare generation.

[ fit an unvented cylinder] this + move it to the attic free up some space.

consider a dual coil cylinder as this would allow solar thermal (whilst oil is relatively cheap now, its unlikely to stay this way in the medium to long term)

Good luck fitting an unvented cylinder on gravity primaries