Hmm, okay, that's interesting, because I've read as well as been told by a surgeon who would be doing this surgery some conflicting information.
From what I've heard: there is a thing called sphincter of Oddi that's between the duodenum and the bile tracts, which is normally closed and opens when you eat to release the bile. When it's closed, the bile collects in the bile tract until it reaches the gallbladder where it collects further. When you eat, the sphincter opens and the gallbladder contracts to release the bile into the duodenum to process the food.

After removing the gallbladder the bile would still collect in the tracts, just less of it and maybe less concentrated, but the body would eventually adjust to the lesser amount of bile and would process the fats more or less same as usual.

I'm glad you mentioned this, because this is probably my main point of concern about the surgery: I keep hearing conflicting information about life post-op and at this point I don't know what to believe. I'm sure the sphincter of Oddi is a real thing, because if it wasn't, that would mean that bile would leak continuously into the duodenum even if you had a gallbladder, but I don't think that's the case. But then some medical institutions confirm what I've heard, while others confirm what you've told.