Remington experienced many manufactoring difficulties, and they never experienced smooth production. Many critical parts are not interchangeable between pistols. The blue was not durable on many of their pistols. Most were likely refinished in the military rebuild project after World War I that ended about 1924 or 1925.
I had a Remington-UMC with only traces of the original finish, but no wear and all matching. The blue finish just turned to a patina. I currently own a Remington-UMC that I believe was initially rejected by the government, and then repaired & refinished before it was accepted. The stampings are faint and exhibit buffing, but the original polishing marks and blue color look correct. The finish is period, but it could have been refinished in the 1920s.
I had one Remington-UMC with a defective thumb safety that I couldn't get to work right, no matter how many parts (safety, hammer, & sear) I switched out. It was probably outside of tolerances when it was made.
While it's unlikely the blue finish on your pistol is original, if it has the correct P barrel proofed on top of the hood, and E marked mainspring housing, then the finish may be original. Have it checked by an expert, and not just someone at a gunshow who saw a picture of a Remington-UMC at some point in the past.