I play the saxophone currently,I've done it for three years.I am picking up interest in the trombone lately.Someone who has experience working with a band said to re-think playing the trombone with braces,because of the putting your mouth inside the mouthpiece you get cut with all the wax and stuff.It wouldn't affect me for starting trombone with braces would it?

It that's the case then,is there some sort of brass alternative that I can do in the meantime?Or could I do trombone?

It's entirely possible to learn to play trombone while you have braces, especially if you are used to them already. You will have to be careful to avoid excess mouthpiece pressure, and the guidance of an experienced private teacher would be very helpful, if not necessary.
The problems will come when the braces come off. You will go through a significant readjustment period. But plenty of people do it and get through it. Chances are you can too.

As Steve said above.

I have 3 or 4 students each year who either get braces on or get braces off. None of my students use wax or other products and all recover their normal range within 4 days of the change, whether it be adding the braces or removing the braces.

Don't use enough pressure to cut or hurt yourself and you will be using just the correct amount. When you get your braces off it's just a matter of getting the lip muscle and skin to elasticize back to where they belong since they have been stretched, but it only takes a few days for that to occur. Just play chromatic scales in the morning and night for one week. You'll be fine.