The problem was that the mower wouldn't start (or even 'click'), and the battery didn't seem to charge up.
I said I'd come back and report.
I have had two great revelations in the past two days:
Revelation 1 Yesterday, looking at this mower battery problem again, having re-read all your good advice, and tried to follow as much as I could, I failed yet again to start the mower. It did cough once, but despite the charger saying "fully charged" (as before), it WOULD NOT go to the charging phase (orange light) for longer than a second. I therefore decided to try and get the battery out once more, and put it on the charger overnight at home -- if it then failed, then it's dead, and I buy a new one.
I have NEVER, for love, money, or swearing very hard, been able to get the battery out of this mower in order to take it home and charge it. It just would NOT, once disconnected, pass through the internal bodywork of the machine, as described in the manual.
Then yesterday when studying the battery extraction problem once more, I realised that the obstruction on the RH side is not a piece of metal superstructure, but the plastic grass chute leading to the collection bin on the back... and the chute comes out!! Pulled it out, got the battery out as easy as pie. (Note -- the chute and bin is an optional extra - not mentioned in the manual.)
Revelation 2 When disconnecting the battery, I thought "Hmm - that positive terminal nut felt a bit loose....". On the way home with the battery, the penny didn't so much drop, as slowly slide down the mechanism of my brain ... I did however charge it overnight (same behaviour: charge has green light on for 8 hours: never an orange).
Put it back on the mower this morning. Did the terminals up TIGHT. And it fired up straight away. 
Hence I have a very red face.
Come all ye young gardeners And 'ark to my song For that which oi sing Won't take very long It may save you time and it may save you strife And it may be a way to impress t'neighbour's wife!
When your mower don't start After winter's long rest You'll likely get out t'charger For to do of its best And if that ol' charger Do say "she be right!" Then just make sure that Your term'nals are *tight*!
Thanks again chaps, and join me in my rejoicing at getting the First Grass Cut done, on a lovely sunny day in Spring!
 Among all the useful and much appreciated advice I received around 11th March, Dave Plowman did in fact say "Assuming a lead acid, and no quiescent drain (unlikely on a lawn mower) there's no reason why it shouldn't retain most of its charge over winter." I failed to add 2+2.
 On my repeated attempts to charge the battery I just stuck the clips on to the terminals without testing the tightness of the actual connections (like what you do on a car).
 The most interesting way to finish that verse. No wives, neighbourly or otherwise, were involved in this saga.