Found the baptism - I see that Arthur was baptised on the same day as his cousin Ann (1 month old in the 1871 census - she and her parents both in Thos and Ann's household).
Camerton, 23 April 1871
Ann, daughter of George Moore (labourer) and Mercy, of Camerton.
Arthur, illegitimate son of Hannah Moore, of Camerton.
His birthdate isn't endorsed, unfortunately, but is likely to be close to the census date (2 April 1871). If you don't already have it, Arthur's birth certificate might tell you where he was born (but it might be as vague as "Camerton"). It's just possible that it would give you some snippet of information as to where Hannah was living at the time - perhaps in service somewhere.
Thank you so much for your suggestion - I think, just think, I might have found her.
There's a Hannah Moore of the right age in service at a farm at Churchstanton for the 1871 census. It's a distance for those times, but not far past Taunton on the Somerset/Devon border, maybe 40 miles from Camerton. Is this feasible?
Also there's an older Thomas Moore in service in the same house. I don't think he's her older brother as I haven't found one of that name, but maybe another relative like a cousin?
It says her birth place was Churchstanton as well, but I can't find a record of a Hannah Moore born there, though if one turns up it rules this Hannah out. It lists everyone of the household as born there, including the servants, so I'm suspicious.
And there's an intriguing bit of oral family history that could be relevant. Whether or not this is my Hannah Moore.
Studying the records it's clear by this time Hannah's father was working in the coal fields and not farming anymore. I always assumed she grew up on a farm but she clearly didn't - a lot of the information on that side of my family tree I've inherited from a cousin who researched it over many years and I'm only now sifting through the evidence with a fresh eye, and at times adjusting interpretation and my previous assumptions.
Now why all this is relevant is the family history passed through Arthur's daughter was we were descended from an Irish farmworker who had travelled to England with his family to escape the famine. That information has been sitting at the back of my mind until now, wondering now and then why there's no recorded ancestor on that side born in Ireland and where the story came from.
The census says the farm was 500 acres "employing 14 men and 3 boys". Farmworkers. Goodness knows who or from where, and she was one of just 2 servant girls.
So just maybe?