What had been happening however was that during the period when the screen seemed to be frozen the iPhone was actually processing the video and saving it, and I had interrupted this process by rebooting the device. I know this because I successfully made a second video on the iPhone while talking with mum in her room - a video of about three-and-a-half minutes' length - and I was able to see that one in the camera roll immediately. By the time I had driven home the first video had appeared in the camera roll, but after uploading it to YouTube I found that it had no sound. So I have a 5-minute video of mum talking to my brother on the iPad but with no sound. Sad.
That's not the only video niggle, however. To sort out the second video niggle, I just now successfully made a broadcast using Periscope. But when I had tried to do the same thing while I was sitting in mum's room in the nursing home Periscope refused to work. The 'Broadcast now' button would not appear and after the program tried to initialise it would just drop out back to the regular iPhone home screen. I guess that the problem in this case was the bandwidth. I do remember checking the phone's signal strength and remarking to myself silently that it looked ok, but it must have been too weak for the broadcast to go ahead, and Periscope had just given up when the required signal strength failed to come through in the program.