I think you need to hang up the call once the other person joins the meeting 

you can record the meeting and you can record the call. 
 If the other person is muted in the meeting and speaking just on the phone then you won’t be able to hear them on the meeting recording, but you will be able to hear them on the call recording.
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You might be thinking … what if the other person doesn’t have a microphone and has joined the meeting without audio?

Ask them to dial into the meeting. In this case the other person would have to join the meetings twice:
- once from their PC/laptop so they can screen share (for example)
- a second time from their phone so they can speak and have their audio recorded by the meeting.

for another person to dial in you can send them the meeting invite (see Copy meeting invite in pic below)

the invite will include a section like this:
 

Or dial:

+44 (20) 34096438 United Kingdom (London)

Access Code / Meeting ID: 422 839 019

I am trying to record what I am doing on my screen during a phone call with a patient, and have the conversation on the video to make training like videos.

in that case if I were you I would mute myself on the meeting.
Record the call either via automatic call recording or call recording on-demand.
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Upload video recording here https://online-video-cutter.com/add-music-to-video
upload call recording there as well.

Export. 

This website was the first google result when searching “add audio to video” and when I gave it a quick test it appears to be easy to use and free.