For travelling long distances on public transport, without it looking like a unicycle, there is one bag you can buy that fits a unicycle and a ton of gear, the North Face extra large Base Camp duffle.
I took it on a plane to New Zealand recently, no problems, no questions asked. It has backpack style straps so you can carry it, although I wouldn’t want to walk more than a couple of miles with them on. I did quite extensive duffle bag research before getting this one, which was the only one I could find that fitted the 29er.
Almost all other bags and suitcases would just about fit a 24, but no way a 29er. They just don’t come that wide or stretch that wide. There are loads of tall ones that’ll fit golf clubs, hockey sticks etc. but they just don’t come wide enough. The downside of this type of bag is that even in this massive bag you have to dismantle it, take the pedals and frame off. It has a big advantage of not saying bike anything and not looking like a bike bag.
I’ve also got a large canvas sack with a rope tie, that I got from army surplus. This fits the 29er with pedals & tyre on, and you can take off the seatpost and bung the seat in it, with your helmet + pads, and it’s not at all a unicycle.
Alternatively, if you’re only doing it a few times, a big rubble bag from a diy store might just do the job, and be a lot more durable than a black bin bag.
I’ve been turned off the tube once, with a coker, and I know at least one person got turned off national express with a 24" muni. Generally with the tube though, if you bung it over your shoulder and go through the normal gates not the big luggage gates with a person on them, you can get away with even a coker. The worst ones for turning you off are virgin trains, I’ve been on them once without a bag, and it was just luck that I didn’t get turned off until where I was going, and even with a bag I had hassle when I carried it on before taking the seat off.
One thing to think about, if you’re not carrying tons of other stuff is that in London, distances are usually pretty small, even from the border of zone 6 to the other side is about 20 miles at most, any tube journey is almost always rideable. In the rush hour, riding a unicycle is probably quicker for most journeys than any alternative except a bike.