Well about the Polish origins of the name anyway.
The name Petropaulowski was in use in English long before the Crimean war.
Searching GoogleBooks for the name shows it is used in an account of Cook’s voyages published in Manchester in 1811, and in the British parliamentary proceedings in 1848 discussing a search to find Franklin’s lost 1845 expedition, as well as other places.
The name in this form also turns up in German and Dutch accounts of expeditions to the Russian Far East in the 1840’s and 1850’s so I guess that it was first transcribed as Petropaulowski, possibly by a German or Dutch merchant, and the name stuck …
But not everyone used it. Using the Google Ngram viewer for Google corpus of English books we see that both names were in use in the first half of the nineteenth century, and crucially Petropaulowski is the most common in the 1850’s
Of course what the ngram viewer does not show is what town Petropavlovsk, or indeed Petropaulowski referred to – there are several in Russia …
… and it wasn’t just a British thing either – David L. Gregg, the US consul in Hawaii (then an independent polity) uses it in his diary entries referring to the comings and goings of the joint Anglo French force.