I am guessing you just have one bathroom and no additional shower room? In which case a combi may do. But first, fill a bucket at the kitchen cold tap, time it, and calculate how many litres per minute it delivers (this is important for a combi). Also measure it at the garden tap if you have one, and report back.
If you can get the immersion heater fixed, and a few electric heaters (and preferably an electric blanket in the end-of winter reductions from Tesco or Aldi), the pressure will be off as soon as the weather improves, and you would find it difficult to get a new boiler fitted any sooner, so don't be rushed.
As expected, all the good heating people will be rushed off their feet at this time of year. Availability, and prices, will improve as soon as it gets warmer.
If by any chance you have to get a new immersion heater, ask for a dual-element Incalloy-sheathed one (they will know what this means) of the maximum length that will fit your cylinder (probablty about 30 inches) and for an Immersion Heater Timer to be fitted, with the clockwork wheel, not an electronic one.
Old plumbers are more skilled in removing old immersion heaters, which are often seized in place, without damaging the cylinder.
The two most popular good-quality mass-market brands of boiler are Worcester-Bosch and Vaillant. I have a Viessmann which IMO is a bit better and has a long guarantee, but they are less common, so be sure to have an Approved Installer as listed on their website. You can get a cheap boiler for about half the price, if you happen to want a cheap boiler.
You can get a combi or a regular (heat only) boiler, it will not make much difference to the cost of installation or running. For some reason, a lot of people think that if you get a new boiler, it has to be a combi. This is not true. As you have an old, open-vented system, it is pretty well essential that you have it powerflushed, and a system filter fitted, before the new boiler is connected to the dirty pipes. Always ask if this is included in any quotations you receive. If not it will cost £hundreds extra. You can get a boiler quote from BG, or any other supplier of Electricity or Gas. Generally speaking it will be more expensive than a local independent, but you can be 100% confident they will not go bust or run off with your money, and if you have a complaint they will (eventually) deal with it.
Combis are more complicated and go wrong more often than regular boilers.
As you have noticed, if you have a hot water cylinder with a working immersion heater, you can still have hot water even if the boiler is broken. This does not apply with combis. It is very upsetting that your immersion heater is not working, because it would have been a boon to you.