I'm glad you had the Anglican minister with you, because the reaction you described was very telling--the police officer had to protect the clerics from even SEEING a Roman collar on a woman. How odd, given that she was wearing what was perfectly legitimate and appropriate under the circumstances. How many kippahs, tallits, and tefillins have they seen at the Vatican without a police escort? How many Sikh turbans? (I have it on authority: they get a pass) How many upavīta? How many robes of Buddhist priests and nuns?  Even the  hijab is acceptable. 

But a priest's collar on a woman isn't. They really do live in an odd world over there. Perhaps the greatest irony is the Pope appealing to women, not as potential candidates to the priesthood, but as reproductive agents to ensure the continuation of male priests. Kind of says it all.