You might try to find the divorce case, which would at least let you know where he was living at the time of the divorce. Since your great-grandmother remarried, that marriage record may indicate the court where the divorce took place.
Dear Ms. Wright,
Thank you for posting your request on History Hub!
Since they were married in England, we suggest that you contact the Gov.UK to request a copy of the marriage certificate and the decree absolute of John Ward and Jean Hoskins Ward.
Once you have the birthdate of John Ward, we suggest that you request a copy of his Official Military Personnel File (OMPF). OMPFs and individual medical reports for enlisted men of the U.S. Army who were separated from the service after October 1912 and prior to 1958 are in the custody of NARA's National Personnel Records Center in St. Louis. In many cases where personnel records were destroyed in the 1973 fire, proof of service can be provided from other records such as morning reports, payrolls, and military orders, and a certificate of military service will be issued. Please complete a GSA Standard Form 180 and mail it to NARA's National Personnel Records Center, (Military Personnel Records), 1 Archives Drive, St. Louis, MO 63138-1002. You also may fax the form to 314-801-9195. For more information see Official Military Personnel Files (OMPF), Archival Records Requests.
Due to the COVID-19 pandemic, the NPRC was closed from March 23 through June 23, 2020. As of June 24th, the NPRC entered into Phase One of a gradual reopening process. The center is currently only servicing emergency requests and will soon expand its service to include time-sensitive requests from veterans for records needed to secure VA home loan guarantees and employment opportunities.
We hope this is helpful. Best of luck with your family research!