I take part in these vigils from time to time. What happens is that a group stands well away from the entrance and says fifteen decades of the Rosary and other prayers and perhaps hymns. Nearer the entrance a counsellor, or a small group of counsellors offer leaflets to those entering or leaving, or to passers by and are available to talk to people. The whole thing is quite peaceful and causes less obstruction than, for example, the people who give out free newspapers outside tube stations. In my experience, I have seen many people stop to talk and it is always a joyful thing if someone changes their mind as a result.
What happens to cause the police attention is that someone from the clinic rings the police to complain about a disturbance, or obstruction or whatever. The police then turn up and feel they have to do something. At the recent case at Twickenham, a car and a van arrived with sirens going, and threatened to arrest anyone who did not stop the counselling.
The Twickenham case seems to have been resolved now. The advice I have been given for others is to inform the police in advance of a vigil, saying exactly what you will be doing. Then if they receive a call, they have some information to go on in making a decision whether or not to attend. Of course, the letter from the Met concerning the Twickenham vigils can now be quoted as an example.