The portion of the railway trail closest to St. George is not really conducive to biking though it can be done. To reach the section you need to travel on local roads for about 3/4 mile before reaching the section. I suppose you could bike on these roads, but they have substantial traffic. Navigating them by foot is also rather challenging even though most of it (not all) have sidewalks. Once you reach the section of the trail itself (just past the oil dockyards), it is mostly flat and you probably could bike it -- but it is also quite short, I am going to say less than 3 miles in its entirety (maybe less than that) before you come to the end (bridge that used to carry the train over the water no longer exists), so not a particularly long ride though the views are quite nice, and the end of the section would be a lovely place for a picnic.
The sections of the trail running form Flatts Village to roughly the Caves (trail ends about 1/3 mile from the caves) are quite nice and quite scenic -- there are a few spots where you would need to walk the bike, but I've seen several bikes on the trail there. You could bike from St. George to the Caves and do these sections in reverse (Caves to Flatts), but keep in mind most of your ride would be on roads. If an experienced biker this would be doable, though I'm no fan of Bermuda's narrow roads....