This is the basic operating principle behind most radiation detectors, but an off-the-shelf fluorescent lamp tube isn't very well-optimized for few-particle detection.
As you have read, a fluorescent lamp works by passing electrical current through a vapor of mercury atoms, which produce a line spectrum with a mixture of visible and ultraviolet light as they de-ionize and relax back towards their ground state.
This light is absorbed by a fluorescent paint on the inside of the bulb, which generally absorbs the ultraviolet photons and re-emits the energy as visible photons.
The biggest issue with using an off-the-shelf fluorescent tube for radiation detection is efficiency. A typical lamp operates with (milli)ampere-scale currents, or of order $10^{19}$ fundamental charges per second.
For comparison, a common radiation unit is the curie, which is roughly $10^{10}$ decays per second --- a billion times slower. A curie is an enormous amount of radioactivity. A typical americium smoke detector has an activity somewhere under a microcurie.
If you could somehow get a smoke detector source inside the tube of a fluorescent lamp, there probably wouldn't be enough activity for you to see the fluorescence with the naked eye. Not a great detector.
You're on a better track with detecting the ionization produced as the radioactive particles lose energy. In fact, this is how those americium smoke detectors work: the radioactive source ionizes a small volume of air, which conducts an electrical current. When smoke enters this air volume, the alpha radiation from the americium captures on the smoke particles, and the ionization/conductivity of the air is reduced. An americium smoke detector is a lack-of-radioactivity alarm.
A way to make your ionization detector more sensitive would be to increase the electric field strength towards the breakdown voltage of the gas in the tube. For instance, you could make the anode of your vapor tube a very thin wire that runs along the length of the tube. The thinner the anode wire, the stronger the electric field nearby.
An ionization event in the strong-field region of the tube might switch the vapor in the tube from insulating to conducting. Since a plasmas have negative electrical resistance, this conducting transition can be engineered to allow as much current as you like to flow briefly through the tube. Your off-the-shelf fluorescent lamp has a "ballast" that prevents this breakdown current from running away, but exploiting breakdown current in this way for single-particle radiation detection was invented by Geiger and Müller.
There are other issues, such as the permeability of the tube to your radioactivity (you need a "thin window" to transmit the strongly-interacting alpha particles, etc.), the probability of radiation passing through your detector without interacting at all, and so on.
You could even detect the mercury emission directly if you had a more sensitive photon detector than your eye.
But at some level these "details" are just engineering concerns; you do have the basic idea of a radiation detector.
