How long will it take to be effective in Tai Chi or Ba-Gua? I have seen demonstrations of them both.?

And I think they are unbelieveable and very effective but I have heard they take a long time to learn, but what is "a long time"?

Really this depends on your instructor. If the instructor shows you how, or knows enough to show you how, you can get a few of the techniques to work in as little time as one year. If you have previous instruction in another martial art then you will probably be able to get it a little quicker than that. Most people starting from scratch it will take two or more years to be able to get the foot work down well enough to employ any of the techniques. These arts are all about the foot work. If you can't get the footwork down properly then all you are doing is a new age health dance. Proper footwork = Amazing power no matter how big or small you are.
If you really want to get involved in these arts try Hsing Yi. I found that it is, for me anyway, the easiest one to learn.
Good Luck

That depedns on the amount of lessons per week. You do a lot of form work before you actually progress onto the applications. If you're really dedicated (say 5 lessons per week) it could still take more than a third of the year before you start the applications.

Ba-Gua can take a while, anywhere from 2-5 year to get the steps correct. but it's well worth it:) Tat chi is about the same.
both relie on stepping motions.I've done both for 3 years and am still leaning new movements.

In my experience I would say that it will likely take years of practice to develop skills that will help significantly in a street fight, especially as most instructors are not really training towards that end. I have seen some amazing things though. Stuff that makes breaking a few bricks (I'm talking about house bricks not baked cinder blocks) look like playing in the sandbox.

Some of the Chinese Old-Timers teach you the forms and the applications. They are one in the same.

My teacher does.

The newer teachers hold back alot of knowledge and many of them don't have the complete art. So they can only teach but so much.