Hi Jackie
The Mr Greenman on the BMD index is possibly a mistake as he was married in Hemel Hempstead, could be a transcription error on the page number.
I have been doing a bit of digging around & I think I may have found your William & Jessie's marriage although not in 1910.
Stick with me & I'll try to explain.
A William Ward married Jessie Frost March qtr 1919 Watford, 3a 1430.
From the info you have already given with regard to Jessie's death her birth would be 1888/89 so I checked out any Jessie's born about then in Bushey. I found a Jessie Gascoigne born c1889 in Bushey on the 1901 census, parents Thomas & Sarah.
I then found what I thought could be her marriage to a George James Frost Sept qtr 1907 Watford 3a 1672.
Hopefully you should be able to find James & Jessie Frost in the 1911.
Thinking perhaps James Frost had died I checked the CWGC site in case he had died during WW1.
http://www.cwgc.org/search-for-war-dead/casualty/750324/FROST,%20JAMES%20GEORGEFingers crossed the link works & you can see that he died in 1916 on the Somme & his wife was Jessie of 7 Rosebery Road Bushey.
Hope this all fits with your info, now we just have to find William Ward born c1874 & your Grandfather Arthur William Ward.
Would you by any chance know if Arthur William had any siblings.
Maddie