Since 1901.
In 1968, the Nobel Foundation accepted a donation from the Swedish central bank to establish a prize in economy, but in hindsight that was a pretty bad idea, and the probability of them accepting future donations to establish prizes in other fields is very slim.
What makes the Nobel prize unique is that almost anyone, even the general public or pioneers in other fields etc can here you received one and be very impressed. You'll generally be met with blank stares if you told anyone not in computing or an enthusiast you'd won a Turing. Even if you then said, "It's the most prestigious award in computing!", it wouldn't hit the same.
Awards like these are basically only really worth their social recognition, so it's no surprise people would still want a Nobel in Computing/Mathematics etc even with Turing/Field etc existing.