watchOS 3 introduced a new workout type, swimming. However, this seems to be available only on the new series 2 Apple Watch.
Is there a way to start a swimming workout on the old Apple Watch (not Series 1, the original from 2015)
No.
Also the original Apple Watch is not made for swimming - it probably won't survive it if you take it swimming every week. The water will simply get in and destroy it.
I would advise buying the new watch for this purpose.