We now have wire transfers and correspondent accounts. The Templars had what we call letters of credit so that people did not have to physically carry their wealth. The Roman empire was pretty big, how would money be transferred? Cash? Would I need a trusted courier or would I have to carry it myself?
The Ancient Romans had an advanced banking and financial system in place. Most of the major financial institutions and transactions that we might use in day-to-day life they had in 200 AD. That’s not to say that everything was as common or that it was open to all people, but various ways to transfer money existed. I don’t know exactly how prosperous the civilian settlement in York was in 200 AD, but I’m assuming there was at least one banker there who had connections in Rome and held accounts there.
Option 1: If you were being paid for military service, or if you were employed in the service of the State in some other way, which is likely in the situation you gave, you could have your salary paid to your wife directly.
Option 2: If the above did not apply to you, you could put your coins in a box and send it on a ship (in the hands of a courier, etc.). But the risks of transport (vecturae periculum) were high, so people avoided this if possible. I know that maritime merchants often insured their goods, but this was expensive. (Many wealthy citizens were creditors and insurers.) I don’t know if individuals could insure goods they wanted to transport. Maritime insurance was a big business, but it was in the form of a type of bottomry.
Option 3: Permutatio was the “transfer of funds from one place to another without any material transportation.” And “its origins probably go back to the first half of the second century BC.” This is similar to a letter of credit / wire transfer / bill of exchange. The idea existed prior to Ancient Rome in Ancient Greece, Ancient Egypt, and other places. The problem is that this was not open to everyone. There were different ways that this could happen, but to give one example:
You’re a citizen and have a deposit account with your local banker in York and have had for some time. You know each other to be trustworthy and reliable. Instead of carrying around cash, he holds almost all of your liquid assets. This local banker has his own deposit account with a banker in Rome, who trusts him. When you want to transfer 100 denarii from York to your wife in Rome, your banker deducts 100 denarii from your local account in York, takes some percentage for his services, then sends a letter to the banker in Rome to transfer the remaining denarii to your bank account in Rome. (Usury rate was around 12%? So we can at least guess that transfers were less than that.) Now your wife can withdraw whatever money she needs and she sends you a letter telling you that she’s happy.
The problem is that not everyone was a citizen or had the reputation with their banker that was as perfect as the above scenario. This was very much a problem of social connections. And not everyone was transferring money between two major cities with specialists for transferring money. Things were simpler if instead you were transferring money or credit from one account in Rome to another in Rome, and more options were open to you. Or if you were transferring public funds (permutatio publica) instead of private funds.
Andreau, Jean. Banking and Business in the Roman World. 1999
Followup: how did these options differ from those before the Roman invasion?
Follow-up: did any "family" actually have the capability to send money between Roman provinces or was that an activity limited to traders and government?
The Templars set up their banking network in the 12th century, nearly 1000 years later. They were more technologically primitive than the ancient Romans in a lot of ways, but they had a key advantage, imported from China via Turkey--unprecedentedly cheap wood-pulp paper, manufactured in bulk with water wheels. The paper alternatives of 200 AD weren't nearly as practical to use for routine financial transactions. Papyrus (made from reeds) was brittle and easily damaged; parchment (animal skin) could be undetectably scraped and altered; both were expensive to produce. For doing business over great distances, metal coins were still the most portable instrument available to a commoner at the time.
Bit of a side question, but wasn't York a kingdom created by Jorvik, a viking? Thus, it would not have existed in 200AD?
A very similar question was previously asked and the answers on that post may provide some avenues for research on this one. See:
https://www.reddit.com/r/AskHistorians/comments/2l9fhu/in_ancient_rome_where_did_the_wealthy_keep/