If someone is wearing a four cornered garment without Kosher Tzitzit (and missing one corner invalidates the whole thing (OC 13:1)), they must either tie on Kosher Tzitzit or take it off. (On Shabbat, of course, the former action is impossible.)
If taking it off right there would cause serious embarrassment, the Rama (OC 13:3) allows one to remain with it on, at least until the next opportunity to take it off privately. Later commentaries there discuss the precise scope of this leniency (and how quickly one must seek such a private place), with greater leniency being offered on Shabbat as one can claim an exemption from the obligation of tying Tzitzit to that garment due to Oneis, and less leniency with our modern Talitot Gedolot which are worn over our clothes where there is less embarrassment in taking them off. A lot probably depends on the circumstances and how embarrassed the wearer feels. I note one suggestion, particularly relevant to the case of a Shul's public Talitot, is to renounce any ownership claim to the garment, thereby exempting you from any obligation to attach Tzitzit to it for at least 30 days (OC 14:3).
In terms of the Bracha, essentially the Bracha said on the invalid Talit was Levatala/invalid unfortunately and does not accomplish or exempt anything. So it's a clean slate moving forward.