Gazz:
What you describe sounds like a whistle meant to be carried aboard sailboats or small motor launches (gasoline engine powered). The piston was likely made up using molded leather piston cups. These are still available for various shallow well water pumps. Molded piston cups made from neoprene, EPDM, and similar are also available in standard sizes, used for piston packing on some types of hydraulic cylinders. A well pump supply firm handling piston pumps (hand or motor driven) should be able to help you with molded piston cups. The leather piston cups will have to be soaking in oil to soften them up for use/sealing.
A 4" diameter pump bore would be about what some hand well water pumps used. The 4" diameter hand air pump you describe was meant to produce relatively high air flowrates to blow the whistle. Even small steam (or air) whistles require a high flowrate to be blown properly (and loud enough to be a signal, if used on a small boat).
I'd also check with a seal/gasket vendor for piston cup packings. We dealt with a seal/gasket vendor in Schenectady, NY known as the Frank Murken Company. We were working on some 1921 hydro turbines/generators and needed piston cups for brake/jacking cylinders. In 1921, these had used molded cup leathers to seal the pistons. Murken got us some molded piston cups made of something like EPDM or similar. These were probably about 6" bore cylinders and were designed to apply brakes to the generator rotor using 120 psig compressed air, and could also be used to jack up the generator rotor (to take the weight off the thrust bearing during layups or maintenance). Jacking used 1500 psig oil. The old cup leathers were worn and would not seal, so we took a sample and some mike readings to Murken. I believe they may be a "Parker" store, a local branch of Parker Hydraulics. I've used the Parker stores for matching up O rings, seals, and other hydraulic system parts. If you find a Parker store in your neck of the woods, you might contact them and see if they can match you up with a modern molded piston cup.
The pump you describe