If I have been attending drill and AT, but I won’t be able to attend BOLC until next year, when can I wear the Army Service Ribbon?

Once you are BOLC complete, which means you are Duty MOSQ/AOCQ, then you can wear the ASR. Reservists and Guard who are Split Option, a break between Basic and AIT, are not authorized to wear the ASR until they graduate from AIT.

You get your ASR once you are fully qualified.

By Regulation you have to wait until you are complete with BOLC.

5–5. Army Service Ribbon
a. The ASR was established by the SECARMY on 10 April 1981 as announced in AGO 1990–15. It is awarded to Soldiers of the U.S. Army for successful completion of initial entry training.
b. Effective 1 August 1981, all members of the Regular Army, ARNG, and USAR in an active Reserve status are eligible for this award. The ribbon may be awarded retroactively to those personnel who completed the required training before 1 August 1981 provided they had an Regular Army status as defined above on or after 1 August 1981.
c. The ASR is authorized for wear upon completion of initial military training. This includes advanced individual train-ing for enlisted Soldiers and branch-specific basic officer leadership course for commissioned officers. For personnel assigned a MOS or AOC based on civilian or other service acquired skills, this ribbon will be authorized upon honorable completion of 4 months active service.