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The job of the person driving the ambulance is to keep moving towards the destination. They coordinate with the dispatcher about what routes are available. The dispatcher is in touch with local PD about road blockages and so forth. The ambulance will, if necessary, change the hospital they intend to go to. A demonstration won't really affect them unless the mother is at the demonstration itself, and the demonstrators personally make it difficult or dangerous for the medics to do their jobs. And no one in a traffic jam or a critical mass is ever held responsible because an ambulance can't get through.
Far more likely in a confrontation between police and demonstrators is that the police start trying to tell the medics what treatment they can and can't provide for people who've been roughed up. In that case, the ambulance people are supposed to advocate for their patients, and insist on proper treatment.