Jan / Johann / John Sigismund / Zigmunt Petrzywalski was born in Liepnik, aka Lipnik, in the Czech Republic but considered himself Austrian.
I have found few details that might be of interest to you. Your ancestor, John Sigismund Petrzywalski was going by the last name Petrziwalsky in the year 1858. According to London Gazette, July 23, 1858 a man named
Johann Von Nepomuk Siegmund Petrziwalsky, of Pall-mall, in the county of Middlesex, applied for a patent for
the invention of improvements in the manufacture of bread, and in the apparatus to be employed therein on the 20th day of July, 1858.
Notice the swapping of the characters "y" and "i" in his last name: Petrzywalski vs. Petrziwalsky in his early years in England. Compare it to the Czech spelling (with diacritics) Petřvalský.
"Johann Von Nepomuk" translates into English as "John of Nepomuk" or "John Nepomucene". The Czech version of this name would spell "Jan Nepomucký" or Jan or Johánek z Pomuka. The confusing thing here is the geographical name "Nepomuk". The place was originally called "Pomuk", but it was later converted to "Nepomuk". And since the prefix "NE" stands for "NO", this have become a source of great puzzling confusion - at least for the uninitiated people like you and I. :-)
Anyway, whether it was Pomuk or Nepomuk, Jan of Nepomuk has become one of early Czech Catholic martyrs (1345 - 1393). He was drowned in the Vltava river at the behest of Wenceslaus the IV, King of the Germans (formally Romans) and Bohemia.
The interesting fact, in relation to your great-great-grandfather, is that he carefully specified whose Jan he had referred to. In the Roman Catholic calendar there are many days devoted to various saints. The name Jan appears there about 40 times in a year. There is a day devoted to Jan Baptista, Jan z Dukli (patron of Poland and Lithuania) and Jan z Nepomuka, etc. My great-grandfather used to stress that his patron was Jan of Dukla, so his name-day was in September - not in June 24, a day devoted to John the Baptist. But he did not have it written down in any document; while your ancestor of two generations before stated it very clearly.
It is worthy to point out that today's Czech people are quite agnostic; nevertheless the Roman Catholic Church is one of the strongest religions there: about 60% Czechs are unaffiliated, there are about 2.1% Protestants, 8.8% unspecified, 3.3% belonging to other religions and 26.8% belonging to the Roman Catholic Church. I cannot pre-judge whether or not this is a strong proof that he was Catholic rather than Jewish, but this is something to think about. Obviously a visit to Lipnik (on Beèvou) cemeteries, especially the two Jewish ones (The Old and the New) should help you very much in this matter.
The London Gazette is quite an interesting source of information about Petrzywalski family. On November 19, 1878 it puts a notice regarding claims against estate of John Sigismund Petrzywalski, Baker and Restaurant Keeper, who died in or about the month of June, 1874, on or before the 16th day of December, The notice mentions his wife Salome as the executrix of the said deceased.
So 16 years later he is already known as Petrzywalski, not Petrziwalsky.
A dozen or so years later, The London Gazette, December 10, 1890 prints the notice about dissolving the partnership in the business of Bakers and Confectionere between Charles Jean Joseph Georges Petrzywalski and Samuel Garner.
But this is not the end of the story: The London Gazette, 6 September, 1927 posts a notice by your Great Uncle Ralph:
Petrzywalski, hereby give notice, that I have renounced and abandoned the name of Ralph Petrzywalski and that I have assumed and intend henceforth on all occasions whatsoever and at all times to sign and use and to be called and known by the name of Ralph Peters, in lieu of and in substitution for my former name of Ralph Petrzywalski; and I also hereby give notice, that such change of name is formally declared and evidenced by a deed poll under my hand and seal, dated the twenty-third day of August, one thousand nine hundred and twenty-seven (...)
RALPH PETERS, formerly Ralph Petrzywalski
london-gazette.co.uk/issues/33309/pages/5784/page.pdf
Well, to complicate the naming issues. There are other variations of the Pet... spelling. Moravske Noviny, year 1868. mentions H. Petrziwalski from Lipnik - a mayor of Lipnik, I think? This time the name has two "I" characters.
Oh, and check this: kdejsme.cz/prijmeni/Pet%C5%99valsk%C3%BD/
This is a current distribution of 81 Petřvalský surnames in entire Czech Republic. They mostly live in Moravia: 13 in Rožnov pod Radhoštěm, 9 in Frýdek - Místek, and only 3 in Lipník nad Beèvou, where your ancestor Jan Zikmund came from.