No Spare Wheel - What to do?

Newish car. No spare wheel. What should I do?

Buy a large aerosol of tyre gloop?

Make sure I have a good breakdown service?

Leave a spre wheel at home - better make that two, the fronts and rears are different sizes?

Start believing in the power of prayer?

Any other suggestions?

Good breakdown cover I'd say!
If you didn't have 2 different sizes to worry about I'd agree with the keeping a spare at home - that way at least after being recovered home (inevitably Saturday/Sunday, with an early start needed on Monday of course) you'd be mobile again straight away.
But in your case you would probably end up with the significant cost of two new wheels and tyres that would probably never get used (or when you do need it, sod's law would dictate it's when you're at the other end of the country...)

I have a "string" type repair kit though I've not used it - I want to try it out on an old tyre first to see what happens. But again it's only useful with a normal puncture rather than anything catastrophic.

What's a 'string' repair kit?

This sort of thing: https://www.tyrebaydirect.com/product-category/tyr...

You use a pointed tool to force the folded over rubber string stuff into the hole, and the string then seals the hole and glues itself in.
It's a "temporary" repair now, but I do have an old Morris Minor manual (when tubeless tyres were a new thing) which states they are alright as a permanent repair.
Personally I'd go for a proper mushroom plug type when possible though

Is there room under the boot floor for a skinny get you home spare wheel, also note you'll need a bag of some sort for a suitable folding jack and wheelbrace, handy if these will fit inside the dish of the wheel.
Should be able to find the right skinny spare used on the usual auction site (make sure it fits before you need it in anger, might not go over large front discs and only fit on the rear, that'll be fun with a flat front tyre on a cold wet night), make sure whatever you use as a wheelbrace that its a 6 sided socket with enough leverage.

I know lots of people don't feel the need for a spare, but blowed if i'm waiting around for a breakdown truck at the most awkward time and place, nor having me pants pulled down paying through the nose for an emergency out of hours tyre purchase.

I have four cars that do not have spare wheels.
If they do not have it as OEM I use a plug-in inflator and tire goop.
Worked fine last week.was on a 2000km run and got a screw in the tire-not repairable and tire was an unusual size so had to go to dealer and run the car on goop for the period , much longer than recommended.

It did leak slowly but a few boosts kept it to normal pressure and I rallied the car anyway.

It is a rare enough event for me-in last year years I have had one blowout and three tires with nails/sharp objects. (Compared to five windshields!!(

I just have a 12v pump in the boot. If a puncture is so severe it can't be managed by pumping it up every half hour, its unlikely any tyre goop would help, in which case its breakdown service time.

In 15yrs driving ive had a slow puncture twice. Never needed recovery for a flat tyre.

People don't intend to drive over a nail..

What about run flats?

Do they work?

What is your expectation of "work" ?

Would they enable you to drive slowly to a safe place ? probably.

Would they enable you to continue as normal ? No.

And of course with runflats TPMS must be working to alert you of the loss of air.