I use a Celebrity X mobility scooter and in the last year it and I have flown from Phoenix, where I live to London/Paris/ on several occasions. I rented a scooter in Japan twice and in Shanghai once.
My advice is to take your own mobility scooter if you can. The rented equipment didn't work too well. I appreciate the hard work of the concierge in Tokyo who found a company but the equipment was not very comfortable and made for smaller people. In Shanghai, amusingly enough, they bought a Mobility scooter for the first time at the 4 star hotel I was staying in and it looked exactly like mine except it had a major difference; it didn't have a brake on the motor..made for an interesting first trip down the hill.
What I've learned is that airlines don't mind working with you if you are friendly and help them. Call and alert them of what you have; that it is a dry cell cell (so they don't have to take out the batteries) and attach the key to the machine and instruct the grounds worker who is going to move it. I ride it all the way through TSA inspection right to the door of the plane and they gate it check it and give it back to me at the other end of the flight. In Paris you can pre-arrange for G7, a taxi company to send a special unit to pick you up.
In London all taxis have small ramps to load your unit into their taxi but in reality you arre better off to schedule the special units
In Japan it is more complex. Contact the concierge at your hotel to find a way to get your scooter into town or use the train from Narita airport.
I would be happy to help anyone find resources. It is challenging but it is possible. I use the public transportation in the US in cities like Portland (fantastic service right from the airport available on their system
Increasingly you'll find this all over the US. Dean Hughson