Possible Original Oyster?

[Bullwinkle]

Hello Everyone. I'm new here and not a collector, but I have in my possession, a watch given to me by my father, beyond which its provenance is unknown. At the time, he indicated it was an early (unmarked) Rolex Oyster, but I'm interested if anyone can provide a better idea of what I have.

The watch is in admittedly overall poor condition, with a well-crazed crystal, and a badly corroded and scratched back. However, it winds, adjusts and runs flawlessly. It is missing the wristband and is quite small, with the ovoid square case measuring 28 mm diameter, and the actual round face (crystal) of 20 mm diameter.

As can be seen by the provided pictures, it is virtually identical to the original "Oyster" of 1926, with the following noted exceptions:
- The face is the original with the Roman numeral dial, lower second dial, and the words "Swiss made" on the bottom sides of the second dial. However it is lacking any branding whatsoever, nor the words "Ultra Prima" which are present in all such pictures I can obtain of the original oysters.
- The hands have a circular embellishment as opposed to the fleur-de-lis of such examples I have seen. Also, they have a very slight colored sheen to them, blue for the minute hand, and magenta for the hour hand.
- The screw down winding stem is of a flattened cylinder as opposed to the more spherical knurled items I observe on original pictures. The flat face of the stem does have raised characters that say "Oyster Patent" and the Swiss cross.

Based on the above, I would suspect that the hands and winding stem may have been replaced at some point in the watch's history, but at least in the case of the stem, I'd assume that it is of genuine (Rolex?) supply, because of its fit and finish.

Based on what I've provided, I'd be very grateful if anyone can provide any more insights into what this watch represents, as well as any advice for me on how I may proceed.

Cheers
JP