"Does Mesalazine have an anticoagulant effect in the same way as Aspirin?" No, I do not believe so. Being chemically-similar but with noted structural differences (as aspirin and mesalamine are) produces dramatically different affects on the body. Mesalamine is 90% topical, so very little of it would affect your mouth.
Second, most UC patients actually clot a lot faster and easier than most of the general population. It involves the enlarged size of our RBCs and their irregular sizes, they block cuts well. If you were to look at your MCV and MCH values in any of your bloodwork you'd be on the high end (or maybe even slightly above normal range) for those things, which is expected of UC patients. In fact, we're much more prone to blood clots from this.
You'll be fine, don't worry