What are the hazards and risks if I keep using a worn rear tyre?

If it’s just the rubber tread which is worn down and thinner: Keep using it. With treadless tires you’ll only suffer from increased susceptibility to punctures. Of course with knobby/treaded tires your grip on soft surfaces will gradually decrease.

If the tire’s structural integrity could be compromised: Don’t use it. It could fail catastrophically at any time. Things like bulges, abrasions of the sidewalls, cuts or large cracks are all bad signs.

Worst case from a worn tyre would be it splitting, and popping off the rim causing the wheel to lock up, and you to probably fall off, or at least damage the wheel.

I managed to get an innertube pinched between the front rim and the tyre on a racing bike, got a few hundred metres down the road and it popped with enough force to completely blow the tyre off the rim, but I managed to coast across the junction I was in and nothing really came of it.

To deal with your hazards:

1. This is possible if you have worn through the grippy layer of rubber entirely on a slick, or are riding a bald rear tyre offroad, however you're far more likely to slip, particularly in an uncontrollable manner due to road conditions. This is more likely to be caused by an old tyre which has hardened than wear.

2. Braking performance is as above really, its probably more likely to lock up, but until you're at the point of locking your wheel there won't be any change. You should be putting most of your braking force through the front anyway.

3. This has too many variables. Is the object glass, flint, thorns, a thumbtack, or have you gotten a pinch flat? Tyres come in many different compounds and tread patterns, and different types of punctures are more or less likely at different levels of inflation, and depends on your inner tubes, whether they are thick or thin butyl rubber, or latex, or even tubeless! Extremely simplified, you'd have a yes/no scenario, with a 2mm object causing a puncture as soon as the tube is less than 2mm from the outside of the tyre.

I cannot think of any country which has a 'bike MOT' as bikes don't have licence plates etc, so it'd be impossible to monitor. Many countries don't even have a car MOT equivalent!

If you aren't sure about a tyre, change it. Decent ones can be found on sale for £10 or less.

As others have said, the biggest risk comes from sidewall failures. Even with the tread worn out there's usually enough (reinforced) rubber to prevent splitting on the tread, but the sidewalls are weaker.

I would say the worst case is for it to blow out while cornering hard. You're likely to lose grip and not get it back again, so slide out and hit the ground. That hurts. Taking this to extremes it would happen while being overtaken very closely and you'd get run over, but that requires multiple simultaneous bad things to happen at the same time that you're doing something (cornering) that's a small proportion of your riding.

A rear tyre blowout in a straight line isn't a big deal in my experience (though I had some warning from the sound of the tube bulging through a damaged sidewall). I have had the rear wheel lock up when riding steadily a few times (derailleur sucked into wheel, broken spoke catching on chain, tyre coming off the bead and wedging against the seatstay)

What's more likely is that you'll get repeated punctures. These aren't inherently hazardous, but can easily leave you cold,wet and exposed to traffic in fading light for which you're not prepared, as you walk home, fix it, or wait for help.

This answer is from the OP, I don't like the fact stack exchange doesn't seem to show it clearly. Also, for now it's just a stub for an answer, written on a mobile, I hope to extend it when time permits.

The question has become a lot less important to me since I started working from home due to covid, but I have been looking into this occasionally.

There doesn't seem to be anything like an MOT for bikes or any detailed rules for roadworthiness. However, case law is a good indication, and I have come across two cases which have some relevance.

Links are below but the bottom line is this: if a cyclist causes any sort of accident, damage or injury, then the condition of the bike will be closely checked. And if the condition of the tyres is questionable to a reasonable inspector, then the cyclist's defence is weakened significantly.

https://www.bbc.co.uk/news/uk-england-hereford-worcester-33236920

http://thecyclingsilk.blogspot.com/2017/08/the-alliston-mis-trial.html?m=1

More reading here: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/685600/cycle-safety-review-report.PDF

The biggest risk is some sort of sudden tire failure, either a "blowout" or the tire actually coming apart.

On a smooth, level roadway with no traffic a simple blowout is usually not too hazardous, but if it happens on a tight turn you could be sent into a nasty slide, if it happens on a fast downhill you could lose control and crash, and if it happens in traffic no telling what could happen.

And if the tire comes apart it's apt to lock the wheel, further compounding things.

(And you don't even want to think about these scenarios on a front tire.)