...you lot...you're a bunch of flippin' animals
Well despite the lack of any tangible assistance
, I sorted meself out: I got Back Up & Restore
on both the phones in question. Backed up all the texts on my old phone, which got put into a specially-made folder on the phone. Likewise the new phone, which had already received a couple texts.
The online tutorials weren't much help for a dinosaur like me (they're in techy-speak, not for the likes of me) but I sussed that if I connected both phones to my computer, I could delve into their files and drag & drop the 5MB or so of texts from the old phone into the relevant folder on the new phone.
Then I had to make Back Up & Restore the default App for messaging on the new phone, and open it up. All the old texts flooded into the messaging folder, and when finished, I reverted to the proper messaging App on the phone.
All my old texts were there!
Uninstalled the App from both phones, sorted
And doing it via my computer avoided any Cloud