OK, I doubt I will be able to come up with anything else that you haven't already considered, but I guess one of the reasons that we use RC is in the hope that a fresh pair of eyes might spot something that we've seen so many times, we no longer notice it. So here's a few thoughts just in case anything might be helpful.
I see Edith Anne was v much a local girl; her family lived a short walk away from Beeley St in one direction and in 1911, Edith was working in a place a similar distance in the opposite direction. I suppose you've ruled out all the possible Edward Stevensons in Sheffield? ---- Including, of course, those who seem to have Edward as a middle name? And, of course, the Stephensons?
You say Edward was of "independent means". So had he inherited something? Have you found any Stephenson wills? Then during the war he was, or became, a mechanical engineer. So I presume he was working on the home front - I wonder, did he try to sign up but be turned down through ill health or something? Maybe he went back to work (had to go back to work) simply because of all the vacancies in industry caused by the disappearance of so many men to the front. Why had he stopped work before then? And when?
Ilkeston was a mining area but as you will know, there were no coal mines on the west of Sheffield. Sharrow was an area of factories and workshops, I believe largely to do with the steel industry. Men working in the steel industry usually had a particular job title specified eg grinder, blade forger etc (as far as I know!) It sounds as though Edward had a different skill set. Where did he learn this? Was he from an engineering family or was he an educated man from a comfortably off background who had learnt a trade in college rather than on the shop floor? The different options would give different pictures of the type of background to search for. I'm picturing someone from a middle class family, who has been educated to skilled professional level, got some private income so has been able to stop working, then goes back to work when the times require. (I could be completely wrong of course!) In this case, maybe not from the class of worker who would normally marry one of the local barmaids (apologies to your granny, no offence intended, but we know there wasn't a lot of fluidity between classes). I wonder if he met her when he went for a drink in the hotel in which she worked! --- That might suggest either that he was local, or that he was a traveller staying in said hostelry -- a salesman??
You say you've looked at the ERs. What about things like Kelly's directories? I have the 1938 one for Sheffield but it sounds as though that would be too late for your search! It does state occupations of the householders, maybe you could get a bit more detail?
School registers for the children? Any info there? (hmm, probably not due to the 100 yr rule)
It's perfectly possible, of course, that Edward did indeed have another family elsewhere and might even have split his time between them. I'm not sure you could rule out someone simply because another woman was having their babies in the same timescale
He wouldn't be the first, if that were to be the case.
These are all just musings, it would be lovely if any were to be of any help but probably not!