Hi Lemmey
There are lots of other things you can check to see how Lemmey's name is listed which might help confirm:
As you say, Death indexes
Burial record
Headstone/Memorial
Notice of Death/Obituary in newspaper
Electoral registers (from 1918+)
Street directories (if she outlived her husband and became head of household)
Baptisms of her children
For most of these, it might require visits to the local studies archives/record office for the area(s) in which she lived after her marriage.
To search for BMD items 1911-1983 you will need to choose the
Complete BMD Index ranges & Images 1837-1983 option
http://www.ancestry.co.uk/search/rectype/vital/freebmd/bmd.aspx and view the indexes page by page as there are hardly any entries on the partial freebmd index post 1911.
No, I don't think you had to show your birth certificate when you married in those days or at least not many registrars/incumbants asked for it if you did.
As regards to Lemmey becoming Emily, I can think of a few possible ideas. Firstly, the way travellers speak isn't always clear to the non-travelling population so what they are saying could quite easily be mishead hence corrupted by a registrar etc.. Some years ago, a late great uncle of mine told me about the "gold needles" on his grandfather's caravan. I really puzzled over these "gold needles" and asked him more than once to clarify - turned out he was saying "golden eagles"

I doubt the clerics came across many occurances of Lemontina/Limentani so unless it was spoken very clearly, quite easy to corrupt

Is there a possibility that the certificate you have could have been mistranscribed from the original register

Or maybe she just became known as Emily - some Emmas and Emilys are affectionately known as "Emmy" which isn't that far from "Lemmey" ie. Lemmey became Emmey

I don't know but something to think about perhaps

Best wishes
Casalguidi