Also, there have been no army records found under the name of Alfred May,
Well no, there wouldn't be. The records on Ancestry are for men discharged or died up to 1921. Alfred May was obviously still serving in 1923, so (as explained in the other thread on Armed Forces!) his records should have survived intact and be available to order by next-of-kin from the MoD.
Info & forms etc. at
http://www.veterans-uk.info/service_records/army.htmlCost £30 and 6-12 months' waiting.
The only thing we can't be 100% certain of is whether 9318 Alfred May is YOUR Alfred May. However, if Alfreda's birth certificate shows her father to be a WO2 (sergeant major) or higher with 1st West Riding in Constantinople (or Gibralt
Ar if you insist; maybe she was born en route but the birth wasn't registered until they reached Turkey - who knows?!), and if the photo kindly posted by Ady in Reply#13 on the other thread resembles any photos of Alfred May that topas may have, then I'd say that was good enough evidence to risk ordering the service record.
Note that, if there's a service number on Alfreda's BC, it won't be 9318. All soldiers were renumbered in 1920, so it will be a 7-digit number starting with 46 for a man in the West Riding Regt.
Adrian