8lpm is very poor, and not enough for a decent combi or an unvented cylinder, or a satisfactory shower.
I think that as well as the big water tank in your loft, you also have a small one, about 18 inches long x 1ft x 1ft. This is very important and probably very muddy. Which is very undesirable. Please try to find it.
Your hot water cylinder is what I would call yellow, so about 30 years old. It is slower to heat than a modern cylinder.
Do you have to turn off your radiators individually in summer, at their valve knobs? I think the HW circulation is by gravity (thermosyphon) but it might be that the pump circulates radiators and cylinder at the same time. This can be fixed fairly easily. Your old plumbing is doubtless full of sludge and sediment, so I recommend having a Magnaclean or similar fitted, now, the next time you have a plumber in. I think you need the 28mm size, which is bigger than usual and may need to be ordered.
First, look at your incoming watermain, at the indoor stopcock (probably under the sink) and the outdoor one (probably by the front gate). It might be lead, iron, copper, black plastic, or blue plastic. Decide if it is at fat as your finger or as fat as a banana, and what it is made of. Do you have a water meter? Is it in the public pavement?
You could speed up your HW heating by (greatest improvement first) converting it to fully pumped, with a cylinder thermostat. Then changing to a more modern cylinder (probably bigger), then adding a timer to control cylinder heating. Better, do all that, at the same time. It will work even if you don't change the boiler yet. If your boiler works you can change it if and when you feel like it, or it breaks and parts are unobtainable. Your new boiler will be about a quarter the size of your old one. You haven't mentioned a reason to change it.
You could improve HW best of all by changing to an unvented cylinder, but it would need to be fully pumped, and you would need twice the incoming water flow. Most likely by digging a trench and laying a new, bigger, plastic water pipe. It is not as difficult or expensive as you think. But plumbers are weedy little fellows with petal-soft hands, frightened of spades, so the trench is usually dug by a burly woman fond of gardening, or a builders labourer.
Insulate all your water pipes, too.