To my knowledge, there is a legal requirement for all cabin crew to be first aid trained. They are generally trained to treat signs and symptoms but to not diagnose. That is a bit of a paradox as the training is to recognise and treat the signs and symptoms of various conditions from heart attacks and stroke through to DVT to renal colic (kidney stones) and emergency child birth.
In a general form, a serious medical scenario is dealt with through a drill which brings together treatment and communication. The most serious and critical scenario being a passenger in cardiac arrest with CPR being performed and the defib (AED) used.
First Aid (or AVMED) actually has an equal weighting to a crew member proficiency to fly as safety and emergency procedures (SEP).
We really appreciate medical assistance, it can be quite daunting dealing with serious "medicals". However, as crew you have to be very aware when a medical professional offers help. There has been a couple of times on my flights when someone with or equivalent of a St.Johns first aid qualification trying to takeover, not realising that actually, the crew are just as well trained.......if not more so.
There has also been instances where a medical professional has assisted but has done something that totally goes against cabin crew training and the crew have had to step in. Case in point is to remember that not all medically trained personnel are specialists and so a General Practitioner (GP) may not be used to dealing with someone having a heart attack. Just an example.
If a medically trained professional advised us to divert, it would be unlikely that crew would decide not to. By the same token, if the cabin crew specifically advised the flight crew to divert, chances are they would. At my carrier we are trained to not specifically ask the medical professional whether we should divert but instead advise them of how long we have until we arrive at our destination. This is to avoid putting pressure on the doctor about whether a diversion is necessary but also there has been instances (albeit rare) where a medic has advised to continue simply because they want to get to where they are going.
I once read a case study at a previous airline where two professionals assisted. A doctor and a nurse. The doctor, whatever their reasons were, advised against diversion but the nurse did. In the end, a diversion was made and it became apparent that was absolutely the right call to make.
Medically trained passengers assist crew thousands of times a year and crew absolutely appreciate their assistance....they can be a lifesaver both in reality and metaphorically. However, we cannot totally handover responsibility to the medical professional for the reasons I have covered.