If you're going to be a mountain biker, you should get some training in swapping tires anyway; flats on the trail are not uncommon. You should be able to trade your knobby for a street-tread tire in a few minutes, with only a single tire tool for dismounting.
Consider checking out thrift shops/yard sales for a cheapie bike with the same size rear wheel set (same number of cogs) as yours. With an entire wheel, you could switch in just a few seconds.
As to the 15 minutes.... Normally, that's a warm-up. Most of the structured trainer workouts I've used are from 45 minutes to an hour, with constant changes of pace/gearing.
However, there's been a lot of research that indicates 15 minute long blocks of fairly intense aerobic exercise can confer benefits. If you only have time for a quick spin....Spin fast.