Well, I can only share my experiences having both UC and having been on blood thinners.
I was diagnosed with UC in the 90's. In 2005, I had a rather significant DVT of my right leg. My UC at that time was somewhat under control. I was on Warfarin for about a year. I didn't have any issues with it during that year, even when I had some mild flares. From the time they took me off of that until this past April, I was on low dose asprin. I never had any issues, even when I began to have a flare in March 2013 that I'm still dealing with today.
In April of this year, I developed another DVT of the right leg. (Caught this one a whole lot earlier. I guess I am trainable.) This time, instead of Warfarin, I was started on Xarelto. I find Xarelto to be much easier. I didn't have to go into the hospital (bed rest at home for 2 days) and there is no need for regular blood testing, like you have on Warfarin. Even when my flare seemed to worsen, but turned out to be c. diff in June, which resulted in a rather hastily scheduled flex sig, I had no issues with the blood thinner. The bleeding that I had from the biopsy during that study was pretty minimal. The thought is that I'll likely be on blood thinners for the long haul.
Of course, if I notice that my bleeding (which is infrequent) would increase, I'd quickly call my primary care doc, and my GI. But in short--my situation with UC and blood thinners has not been much of a concern. (Sure beats the risk of another DVT or developing a pulmonary embolism.) Hope that helps a bit.