Are they fluorescent of some sort, if so, that's not too surprising.

No, they are incandescent, just ordinary can lights, one over the tub is covered the other in the ceiling bare.

Since you imply that the bulbs are identical (i.e., ordinary incandescent), why not swap bulbs and see what happens?

If you get the same result, you will have helped isolate what is going on here.

You know, come to think of it they may not be identical. It is possible the covered bulb over the tub is florescent. Since I can't see it because of the cover I don't know. I will look at that first. Maybe that is the answer. The electrician supplied the bulbs. Now, how do I get the cover off?

Call the electrician and ask them this question. They will be able to tell you what kind of bulb and you don't have to undo the cover or worry anymore.

We have two lights on a switch that do the exact same thing, first one is incandescent and second one is a fluorescent twist bulb.

I just checked. It IS florescent! So that delay is normal with florescent ? I'm relieved! Thanks!

Yeah, fluorescents can take anywhere from zero to a second to start up. They even get slower as they get older.