PeterJGregory

Thank you for your helpful link to wordpress.com support.

Yes, the new site is with WordPress.com.
My old wordpress site is hosted elsewhere.
I have tried asking WordPress.com support a couple of times about this (sending emails, as requested), but have not yet received a reply.
Perhaps I need to wait a bit longer (or persist)!

I don’t think this is possible as I am unable to install any plugins in my ‘new-name.wordpress.com’ site without a paid upgrade (to a Business plan, I think).
Have I misunderstood you?

The mobile theme for the Big Brother theme doesn’t appear to be working correctly
When I enable it under Mobile Options I find that my menu is replaced by links to Facebook and Twitter.

Not heard back from you yet (having sent the further information you asked for) so thought I would try using the Twenty Sixteen theme.
When I did a live preview of this theme (not the child theme) I noticed that the png images were not shown (and neither was the header image). But when I activated it, they all were shown.
I then created a Twenty Sixteen child theme, and found that the live preview again didn’t show the png images.
So I am wondering if this is just a function of the live preview and that the images would be shown when the child theme is saved and activated.

Could you please give me your thoughts on this.

So far I have only added png images to the one page of my website, so it would not take much effort to add them again should they fail to appear when the child theme is activated. If I saved and activated the child theme for Twenty Twelve then found that the images were still not there, would it cause later problems if I added them again?

I have now sent you more information as requested