Certainly the Royal Artillery, but from 1939 the British Army replaced the WWI style Service Dress with the “short tunic” battle dress uniform, so I cannot confirm a date for you. All I can say is wearing collar dogs was suspended during WWI, so it's later than the mid 1920's.
Pre or early WWII see if there are any surviving letters home, his discharge certificate, pay book or the box his WWII medals were issued in; any of those could have his Army service number on.
His records, including WWII campaign medal entitlement will still be with the MoD see http://www.veterans-uk.info/service_records/service_records....
An application should ideally have his army service number, failing that his National Insurance Number, any application with just his regiment and date of birth risks simply being a £30 donation to HMG.
Re RAF records
You said, “now to find the cheapest option to see his record.”
If you live reasonably close to Kew, then to experience the thrill of personally finding a touching (albeit through gloves) original documents of ones ancestors then a personal visit to Kew should be considered.
If travelling to and from Kew in a reasonable time and cost is not an option then you have the option of using the NA’s Staff to extract and copy the record (now £20 per ¼ hour) or use a researcher. Two options to find one:
Google for - military research on line – one of the top listed should a company that offers fixed price packages for many types of record and a very reasonable hourly rate.
Alternatively from http://www.nationalarchives.gov.uk/
select Records from the top menu bar, then scroll down Paying for Research, and look at the options of using the NA’s staff or an independent researcher. In the list of independent researchers you will find many with web site and email addresses so contact them, tell them what you know and request an estimate; from there it’s your choice.
All will be plus VAT & copies but beware plus expenses as if they are regular users of the NA at Kew, expenses should be included in their prices.