It appears that my problem with this site was not a one-off. Quite a few of the sites that I have hosted with TSOhost were affected by the server switch. For some reason during the migration the level of PHP that was chosen was 5.4.45, with memory_limit set to 25M. … but post_max_size and upload_max_filesize set to 256M.
For all the other PHP versions above 5.2 the memory_limit is set to 256M. I have no idea why they chose PHP 5.4.45, being out of support for quite a while now.
This effed up quite a few of my sites, especially those running JetPack with certain modules activated.
I’ve written to the hosting company. But they don’t seem to know how to apologise. One more nail in the coffin.
This site is currently running: WordPress 4.7.3. PHP: 7.0.16. Memory limit: 128M