Generic word for uni-, bi-, tricycle, etc.?

Solution 1:

Pedal cycle.

At least in British English, it is a recognised generic term, sometimes in a formal or legal context such as insurance

Note that the examples of what is covered don't explicitly include numbers of wheels other than two, the term itself is not limited to 2 wheelers.

Solution 2:

A velocipede:

Velocipede (/vəˈlɒsəpiːd/; Latin for "fast foot") is a human-powered land vehicle with one or more wheels. The most common type of velocipede today is the bicycle.

The term "velocipede" is today, however, mainly used as a collective term for the different forerunners of the monowheel, the unicycle, the bicycle, the dicycle, the tricycle and the quadracycle developed between 1817 and 1880.

[Wikipedia.org]

Note that this is unlikely to be understood by a wide audience. I knew the word existed, but I didn't know its exact definition until I researched it for this answer. I had suspected it was just a synonym for bicycle, and was quite happy to find out it was in fact its hypernym.

Solution 3:

A cycle

4.
a bicycle, motorcycle, tricycle, etc.
_{-Dictionary.com}

Note: This was posted before the exclusion of 'motorcycle'.

Solution 4:

Re:

Is there a generic word for bicycle-like vehicles that may have 1, 2, 3, 4 or more wheels?

Technically, yes, see AndyT's answer of "Velocipede".

But practically, no there isn't.
Not every concept has, or needs a word in English. The proof is in the fact that Velocipede never caught on (nor any word like it):

Practically, the word "Bike" is close and and most people will get your meaning.

"Bicycle" shops sell unicycles, bicycles, tricycles (AKA "trikes"), and quadracycles (AKA "Car-Bike" or "Twin bike", among other aliases).

Of course, "Bike" can also mean motorbike. Again, this hints that there is no real need for a unique term.

In terms of: ease of use, (human) power requirements, safety, efficiency, stowability, and support infrastructure, there are huge differences between 1, 2, 3, and 4 wheeled, human-powered vehicles (HPV). Likewise, 5 or more wheels would almost never be a good engineering/ergonomic solution for an HPV**.

So, it makes a great deal of sense to keep separate terms for such devices. They are markedly different experiences from the rider and owner perspectives.

** HPV does not answer the question since it also applies to vast hoards of watercraft, aircraft, etc.