GDoS suggest a possible origin from medieval French roussin:
(also rawser, razzer, rosser, roz)
[? Rom. roozlo, strong or roast, a villain; B&L suggest rousse, roussin, a policeman (from Medieval. Fr. roussin, a warhorse or hunter)]
a police officer; also attrib.
1888 [UK] Sporting times 26 May n.p.: Up walks a rozzer and buckles me tight [B&L].
1956 [UK] ‘Charles Raven’ Und. Nights 148: The Surrey rozzers pride themselves on the efficiency of their cordon system.
World Wide Words has other suggestions, none of which appears to be conclusive:
A common supposition is that it comes from Hebrew khazeer or Yiddish chazer, a pig, but this is almost certainly a guess derived from the 1960s slang term.
Yet another candidate is the Romany ruzalō, strong. Some point to
roosher, contemporary with rozzer, which is listed in Farmer and
Henley’s Slang and its Analogues of 1903, but that merely transfers
the problem to another word of which we know nothing.
None of these have any direct evidence to support them. Once again, it’s “origin unknown”, I’m afraid.