The answer to this may help others, I hope.
It is not ProcessWire (I didn't think it was but I was hopeful someone else might have had some related knowledge that might have helped hence bothering you guys here). I had already spoken to the hoster in this case to work out which php.ini file was to be edited.
The answer is interesting and easy: as I was running a website with FastCGI the php.ini file just above the httpdocs location was the one to edit, not the overall one often found in /etc/php.ini. All good, I edited it. But the problem remained.
So a call to (mt), the hosters for this site, and they quickly found and fixed a one-time file that stops FastCGI working with files greater than 128KB. They upped the limit for this file to 1GB and so now this site and any others that use FastCGI will practically be controlled by the settings in their php.ini files.
Result: all is working beautifully
I hope this is helpful to someone else and thanks again PW people for this superb product. Cheers, -Alan