I was going to move my Legacy to a more legacy friendly world, since we can do that now, but I had already started to time travel before I thought better of it. If I DO move my family to another world, will everything I've done in the future be lost? I know that you lose all friends, and family ties, but I'm not sure how it would effect time travel. I would hate to lose all the progress, even though it is very little.

I moved my sim from Isla Paradiso to Hidden Springs because I had constant meteor showers in the normal and utopian futures and wanted a fresh start. Well that didn't happen, the meteors followed me. The changes I had made to the community house ( a Bonehilda coffin and some bunkbeds ) were there as was the meteor damage. My legacy statue was still in the park. Apart from Emit my sim wasn't friends with anyone previously, her friendship with him was unchanged. The descendants were different but still looked like her and she didn't know them. So from my experience whatever progress you have made will be kept.
A quick warning: Make a save before moving and back it up. Then rename your new save as something different to be on the safe side. The game prompted me to rename my save when I started the new town which I did, but it didn't take. I called it Hidden Springs Zain (Zain being my sim) but it was not in saves, they just overwrote the previous save Oasis Landing Zain. It really didn't matter as this is just a test game but I can see this bug would be annoying if you are playing a precious game.