**Editing to say that I am 100% sure you can swap batteries between the First Gen and Series 1 Apple watches because I’ve successfully done it countless times.**
I'm pretty confident that you can use a battery from a Series 1 Watch in an Original Apple Watch. The internals between the Original and the newer Series 1 are almost identical. (The processor was about the only change.) In fact, if you order a replacement battery from ifixit.com, the batteries are listed as being compatible with either Watch. Also, the details about the used OEM batteries listing even says that they are taken from either an Original or a Series 1 Watch.
Apple Watch (38 mm Original) Replacement Battery
Apple Watch (38 mm Original) OEM Battery
Series 2 is different though and cannot be used in the earlier models.