This is one of the reasons money exists
Earth recently had this problem. From the dawn of monetary transfer until the invention of radio and the telegraph, information about a financial transaction required days to years to travel everywhere it needed to go.
So people would take their gold with them. Except gold is really heavy and costly to transfer.
So people converted gold to something more valuable: like jewels. Still too heavy for really big transactions, though. And jewels are somewhat limited. You can't have just any jewel, they need to be cut! That means they're not always quick to come by.
So some idiot thought to himself, "you know, if I simply write 'one bazzillion dollars' on this sheet of paper and have it verified by people with authority and a good reputation, that would be really easy to both carry and hide!" Voilà, money.
But even that's inconvenient. Money can be stolen. It can be destroyed by fire! It can be eaten by the dang camels! And so Man created...
Poor communication means localization of financial activities
When you're talking about settling a new system, what financial needs do you have?
Your people need something with which to do business one with another. But that doesn't require the rest of the galaxy. That simply requires a local mint and the signatures of authoritative and respectable people.1
But your colony needs to do business with the greater universe. It needs to import and export goods!
And that means businesses and banks manipulating credit to get the job done, because sometimes the physical cash simply doesn't exist to make it happen and it would take too long to get it in place.
But if you think about it, just like 2,000 years ago when every individual couldn't do business world-wide but had to use all kinds of middle-men, your settlement won't be doing business with someone so far away it takes too long to get the requested materials. They'll be dealing with local brokers.
So your financial institution doesn't actually need the ability to communicate across vast distances regularly. It needs to be locally stable with galactic coordination and accounting.2
OK, but how do I coordinate on a galactic basis?
The same way we did not too long ago. The U.S. once minted \$100,000 bills that were only used for value transfers between large, official institutions. You'll be doing the same thing, just digitally. When your system-banks3 need more liquidity (or are rolling in wealth and need to transfer some of that snuff to another system-bank that needs liquidity), they roll out the proverbial \$100,000 bill — an electronic transfer of astronomical proportions (hah). It's deducted from one bank and not credited until its arrival at another, no matter how long that takes.4
Which means the "central bank" isn't a bank so much as it is a clearing house — and a policy maker
Which sounds a lot like the U.S. Federal Reserve, doesn't it? It would likely stand outside government oversight, too. And if we're really lucky, it's run by a combination of the Rothschild family and the descendants of the Knights Templar.
1 When's the last time you even LOOKED at the signatures on any form of currency carrying one? The trust hasn't been in the individuals for centuries. Maybe even millennia. We trust the institutions (like governments) that issue the currency. But those dang signatures are still there. Old habits die hard.
2 And thus audits were born. Because those respectable people aren't always as respectable as one might think....
3 Which sounds a lot like "system lords" and should be treated with just as much fear and loathing.
4 Which means the transmission can be intercepted! Woo-hoo! Intergalactic train robbery! The new Manga to be available on Japanese bookstore shelves next week!