Single, no knowledge of roofs and feel I might make a financial mistake, please help?

Does it actually need doing or could the cracked tiles be replaced and any holes in the underfelt/boarding be repaired /patched up.
Have you been in the loft and inspected if water is or was getting onto the beams/Roof framework as making sure those “A frames” are kept dry and don’t rot will save you thousands in the long run.
Water runs in a downward direction
If water comes in at a certain point, where is it going to? Is it running across a beam or going straight down onto the floor and getting between gaps to sit somewhere in the ceiling of the floor below.
Ultimately building work is very logical and operates on a If it ain’t really really broken don’t fix whilst making sure you keep on top of the little things (like the cracked tiles) to prevent having to fix the big things.

why wouldn’t they want the work?
You said: "don’t have any slots to do the job until October". They don't need the job now but might consider slotting it in if you are willing to pay silly money.
When is best to get a roof done?
When it's not raining! Rain slows down the job.
The lowest rainfall (assuming UK) is between March and May. The lowest number of wet days (i.e. the day is likely to be dry) per month is Apr-Jun. The wet season is October and onwards.

Quoted 22k. Very thorough, don’t have any slots to do the job until October so obviously busy.
Do be aware of the "go away" quotation. They won't say "go away" but will make the job appear so expensive that you will go away of your own accord. Maybe get a fourth quotation?

When are you thinking of getting the work done? Presumably not during rainy Spring. Try to make a decision soon so you can book the work to be done at a time that suits you. Make sure that the quotation holds good over that time-span.

@Rooooof do you have a facebook group for your local area? I find these so helpful for reviews for this sort of thing as it's real people rather than some unknown online

Please be aware that you will need Building Regulations approval if you are replacing more than 25% of your roof.

How long have you had the house for and what did the survey say?

A tradesman has to have a waste carriers licence and take any trade waste to a transfer station, where it will be weighed and he will pay the fee, he is not allowed to take it to the tip or put it into the general household waste stream. Most councils now require vans to have a permit at the tip/recycling centre and have strict limits on the amount of building waste they will accept. My local one will not take asbestos, plaster or plasterboard, rubble/bricks is limited to one quite small sack.

OP I would suggest asking neighbours, friends and local Facebook community groups for recommendations.

What private individuals can legally do, is bag the stuff up themselves and take it themselves to a designated disposal centre. How to do that is easy, not complicated. You wrote in your OP

bag up the few asbestos cement sheets and take them away for us

which is consistent with what I understand he can do and is safe enough to do. If you're concerned he should have a license, you could phone the council to check on the rules for trade about licenses for trade persons. Without a required license he might be tempted to put the bags in ordinary waste stream and I wouldn't like that either. I'm not pretending knowledge I don't have, just saying that

bag up the few asbestos cement sheets and take them away for us

sounds reasonable to me.

I have disposed of asbestos legally. Double bagged it whilst wear PPE and booked it into my local tip. They take up to 10 bags for free by appointment. Please don't assume he's a cowboy. Second time I needed to dispose of more and just booked an asbestos sealed skip. Same double bagging and PPE then they take it away. It's not necessarily illegal to handle asbestos and there's a huge difference in the various types. Both times the work was part of larger works and signed off by building control.
I second the opinion that if the roof isn't leaking then leave it alone unless you are planning to sell soon. Also if it's just some cracked tiles then no need for whole new roof.

I would get a surveyor to check the roof and tell you honestly what is needed . It could be worth his cost because you would be able to make more of an informed choice .

We had a leek in my bedroom, the flashing ( piece of lead on roof between properties) had dropped down into the gutter. my bedroom celing was wet. This was a simple repair and did not require a whole new roof.

I put a lot of store in what someone’s reputation is locally so i wouldn’t discount no 1. I have no idea about asbestos disposal though.

I ve never asked anyone about vat registration though so that wouldn’t have crossed my mind either.

Is the roof leeking now?
You need to know specifically what is wrong with the roof. You need photographs of all the damage.
Go on youtube once you know specifically whats wrone, to see how experts would treat the problem.
i got drone pictures of my roof which was very helpful, because of the picture i could identify the problem myself.
Its in a bad condition but will hold up for a few years isnt specific enough to part with 10s of thousands of pounds.
Knowledge is power.

Some types of asbestos, it's legal to dispose of yourself, I don't think No. 1 is a cowboy in what he describes he'll do.

The middle quote if you take off the VAT is £12k, so not massively more expensive than the Bodgit & Scarper Roofing Co, especially if they are using a properly licenced waste carrier, rather than taking it away and before dumping it in a gateway somewhere. Do the quotes include scaffolding

In general if you have an ageing house, it can be more cost effective to do new works such as a new roof, rather than patch repairs which may need re-doing in a few years time.

However, whether it’s cost effective for you depends on lots of things…

How big is your house slush fund? Will you be able to save more into it regularly?

What else will you want/ need to do with your savings?

Are there other issues coming up with your house? I’ve recently had issues arising from the intense storms we have more frequently, compared to when I bought my house 20+ years ago.

How long do you intend staying in your house?

So, for my recent roof experience, house was built in 79, age- related issues are beginning to show up, I need lots of things doing including a new kitchen and a general upgrade, refresh and redecorate, none of which I can do myself because of disability. I have an overall budget of up to 50K (which is partly spoken for for disability related equipment). Roof has leaked a bit in intense storms over the last 5 years - inconvenient because the water comes through a light fitting and then that bit of the circuit needs isolating by an electrician. I intend staying in the house another 5-10 years but could well be longer. My quote for repairs from the good firm was about £1600 but they couldn’t offer a guarantee due to the nature of the work. New roof with 20 years guarantee was from memory 11 or 12K. For complete peace of mind and probably no roof hassles for the foreseeable future, I would have gone for the new roof, but due to a close relative’s very recent terminal diagnosis, I decided on the repair. It may well be that in 5 or 10 years time my perspective will have shifted, the roof leaks, and I decide on a new roof, but that’s houses….

As it’s a major expense and I think you can discount the cheapest quote, 2 quotes is not many. I would get some more quotes and sit on this decision for a while, the right action for you will emerge.

Do not touch the cheap guy.
It is illegal to "just bag up" asbestos & the consequence of exposure to asbestos are dire, life limiting & can lead to a nasty death.
If its not disposed of properly by licensed operatives you will be exposed to asbestos particles.

Why wait?
The cost of building materials is going up far more than inflation so it will cost more in 2-3.years. I dont know about you but everything time we have bad weather id be awake worrying about the ancient food.

Not sure what makes you think the whole roof needs replacing. Just repair the cracked tiles.

A couple of cracked tiles? If so, its a few hundred quid for them replacing. What makes you think the whole roof needs changing?