No, no, no! You must never give in!
Yes, it's very possible your G nan dropped initials - or even whole names. It's also possible that someone else was doing the 'dropping'. In the case of census records the original household form was transcribed into the enumeration book and these don't always match. The book has been further transcribed to digital format by companies like Ancestry and again, they don't always match. People didn't always give their regitered name and simply wrote down the names in every day use and lastly the birth reister birth certificates) doesn't always match the baptismal register. Forms didn't always have room for all names and the writer sometimes abbreviated to an initial or simply missed out a name. Going back a hundred years or so, parents waited much longer to baptise their children than they do today so many had time to think and the baby got an extra name at the baptism which doesn't appear on the birth certificate, So, you need to think of all those possibilities, variations, additions and transcribing errors when you research.
When I first looked at your query regarding Margaret Alice Walker I couldn't find anybody with that name: I only found Margaret Jane Walker or Alice Jane Walker with that birth year and birth place. Margaret Alice Jane showed up first in baptismal records and marriage records for the appropriate year and place. So it's possible your G nan got an extra baptismal name. Two of them had parents named Henry and Dorothy (but two completely different couples) and one of them had parents named Michael and Mary J. I chose to follow one of the henrys because of the Blyth connection.
I can't say with certainty if Elijah on the marriage certificate is from Haltwhistle. Even the marriage certificate wouldn't give that information - only his residence at the time of the marriage. However, it would give the name of Margaret Alice J's father which would solve the query of her parents. Myself I'm 95% certain that it is the same Elijah because so many other things fit in. Had he simply been called Elijah Smith I wouldn't be sure but how many Elijah 'Wilkinson' Smiths married a much younger girl named Margaret AJ Walker in that registration district in the early 1920s? I can't find any. Likewise, I am 95% certain that it is the same Margaret A J living with James Waldock in 1939.
Two questions:
Do you know if Margaret Alice and Elijah had any children? There are some interesting Smith births (mother's maiden name Walker) in the early 20s.
Margaret Alice's twin girls - did they have the surname Smith (as Margaret doesn't seem to have married Mr Waldock)
Divorce, way back, then was relatively new (1858, first divorce law) and an expensive luxury affordable only to the rich. So people chose to go their own way and "live over the brush" with someone else. I'm not sure, but I also think that it was only men who could apply for a divorce. Some, almost certainly refrained deliberately to prevent their estranged wife from re-marrying. You hint that shame or scandal may have been involved. It needn't necessarily be so. Perhaps Elijah wasn't kind and Margaret left him.
Whatever the case may have been, I think your G nan must have been a very strong woman. It can't have been easy remaining in Bedlington with all the wagging tongues. Her early life was either exciting or tragic and it's up to you to get to the bottom of the mystery. If I can be of any help feel free to ask.