When did Remington change their headstamp from Rem-umc to R-P?

No earlier than 1934, when Remington purchased the Peters Cartridge Company.

__________________
"I also cook."

Sometime in the late 1950s-mid 1960s. They were still selling ammunition under both the Remington and Peters until that era, although toward the end of Peters marketing both brands were head-stamped R-P. Probably 1957-59 was where this occurred.