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And I solved it myself soon after posting (as is often the way)!
It was because I had set Complianz to “Respect Do Not Track and Global Privacy Control”.
I was only seeing the tick-boxes in section 7.1 when logged in to WordPress. When I logged out I got the same message as on the mobile devices, namely “You have loaded the Cookie Policy without JavaScript support.” However, when I tried in Chrome I got a different message in 7.1 – “We have received a privacy signal from your browser. For this reason we have set your privacy settings on this website to strictly necessary. If you want to have full functionality, please consider excluding https://kenyachildrencentres.com from your privacy settings.”
That gave the clue.