Is it correct to write a sentence like so?...

I will be available on Tuesday: time one at 11:00 a.m.; time two at 11:15 a.m.; time three at 11:30 a.m.

It's not bad.

I would prefer (using your words):

I will be available three times on Tuesday: time one at 11:00 a.m., time two at 11:15 a.m., and time three at 11:30 a.m.

You only need to use semicolons if your list items contain commas, which they don't. That said, your semicolons are probably useful because of the number of full stops (periods) in your list items.

So, this is acceptable:

I will be available three times on Tuesday: time one at 11:00 a.m.; time two at 11:15 a.m.; and time three at 11:30 a.m.