What is the URL that appears on the screenshot that says " "Upgrade completed successfully"?

You may have upgraded b2evolution in a different directory.

That's a good thought as I don't really know what I am doing. The image of the URL is gone now so I can't answer the question unfortunately. I do know that the many 'depreciated' messages that were appearing are no longer present, so I must have done something right. But the version on the Dashboard is still showing 2.4.1 and the UI does not seem to have changed at all. Further evidence is that the space I am using for the blog is much more now.

I reverted to a backup copy and tried again. Still no joy :-( I will attach a screenshot of the error message that displayed. What should my next step be please ?

I looks like you have not correctly uploaded the b2evolution files and that you haven't uploaded them to the correct location.

When you say not correctly uploaded, do you mean that some are not copied ? On the server the compressed installation file is 5.8MB and when uncompressed it is 21.6 MB - should there be more ?

What is the correct location ? I have the installation directory at the same level as the blog like this:
/home/public_html/index.html
/home/public_html/blog
/home/public_html/b2evolution

Thanks for helping.

Any help appreciated.

The location can be different on each server. Now you have uploaded your own stuff, probably in the wrong location, it's difficult to recognize the right location.

Check this out to determine the right location:
http://b2evolution.net/man/installation-upgrade/faq-install-upgrade/where-to-upload-the-files-for-upgrade