What are the dangers of a DIY fuse in a multimeter?

It's the same with disabling or bypassing any safety device, I believe you could be completely safe, but what if someone else picks it up and uses it? The danger is of arc flash, of course.

Perhaps if you could mark it (and cover the model/Cat/IEC markings that would lead one to believe that it's safe to use on 600VAC? "Do not use on mains". Not sure if that is legally necessary or sufficient in the UK, but it might reduce the possibility of injury.

Here's what's left of a multimeter that was involved in an accident that killed two people.

Evaluation of the meter circuit showed that it used a small glass 8AG fuse rated 0.5A at 250V for circuit protection on some functions. According to Underwriters Laboratories, the interrupting capacity of this style of fuse is only 35A at 250V. It has no specified interrupting rating above 250V. An estimate of the fault current through the meter shows that it could have been from several hundred to as much as 1,000A at 277V

The whole story is here. Note that the circuit was not even an industrial circuit, and "only" 277VAC phase-to-ground, but 480V phase-to-phase. The available fault current was not small.

I once tested some 5A/250VAC rated ordinary 5x20mm fuses on a light industrial 240V circuit (50A circuit). Almost every time they arced from end cap to end cap, and the glass tube literally exploded. Molten metal was found to have solidified in a layer on the glass shards, so there was a cloud of it after the tube ruptured. A plastic housing would have contained the shards, but anyone foolish enough to be closely observing without a face shield or safety glasses could have been injured or blinded. Interrupting (current) capacity is an important factor, and it's not marked on fuses generally.

Wow, there's a huge price range on that fuse- I see everything from $5 to $36.

The risk of fitting a fuse with inadequate breaking capacity is that it fails to interrupt the fault current (see note below) and instead arcs over. If you are working on a high current circuit at the time that can mean your multimeter exploding in your face causing serious injury or in extreme cases even death.

Glass fuses are especially bad. They have very low breaking capacity, often as low as tens of amps. A cheap ceramic HRC fuse would be much better than a glass one but would still not provide as much protection as the correct specified fuse.

Interestingly, looking at Farnell (I'm assuming from the pound sign in your question that you are in the UK), it seems that the Bussman branded fuse is £12.15 (ex VAT) BUT they seem to have a suitable fuse sold as a fluke spare part for only £3.70 (again, ex VAT). At that price. there is no reason not to fit the correct fuse.

Note: Fault in this context can include both actual faults in the circuit you are working on and user screwups like plugging the probes into the wrong sockets.