My bike was stolen and I could not afford to get a replacement in the quality I wanted. I could have bought and older model which needed work. So I picked up a higher end electric bicycle to get around the urban area. It is great fun and works well for trips that are under 40km from a railway station, but its much to slow for 1000km day trip. But I can still do a 1000km fortnight trip on the bike if I want to.
I just use plywood ukes for touring on long trips on bicycles and when I had my motorbike. Just in a simple bag, you could use a pillow case if you lost the soft case. Or even without a case, just tied on top of the luggage. In the sun mostly, and a little bit of rain. The beauty of leaving the (soft) case packed away is that when you stop for lunch, the uke is on top, easy to access and play while the water boils. As long as it does not interfere with your vision, you can tie a uke on top, where ever you choose. It is very light and if you work it out, you will not need a high tension strap to hold it on safely, maybe just an octopus strap. Another option for a shorter trip is to hang it off your back, which might be dangerous if you do fall off. You really do not need a hard case on a bicycle or motorbike, you just need to carefully tie the uke on top of the luggage, and you need to bring the right uke.
If you are going to a festival or job or event where you have a playing part, maybe then you would pack a nicer uke inside a hard case and strap it onto the bike. But you can find plywood ukes which will do the job for most other two wheel adventures. If you are really keen, you could even commission a custom plywood uke which has some stronger attachment points so you can tie it on. Maybe you could order a nice little plywood soprano with a painted finish to match the bike, and with stronger strap pins fitted and a headstock with an attachment point so you can just tie it on top? Give everyone a surprise when they think you have a painted over $30 Mahalo and it has an amazing tone and is easy to play, but its actually a custom uke from a well known maker?