Given that our mental mindset does not change, trading would start immediately (on day 1).
I need that and I have this, can we swap?
It will not take long before someone figures out that he can start collecting objects of type X, because X makes good trading stuff. Take cigarettes for example, you see these being used for payments everywhere. My estimate is a couple of weeks at most, before X will be recognized as good/common trading material on a regional/national scale.
Valuable metals would serve the same goal, and maybe other physical objects too (anything with a certain scarcity/desirability will do). The diversity would be less than all the payments systems we know historically, because we now live in global world. So again, weeks to maybe months before we have a number of goods that are used as payment tokens across large regions or around the world.
The next essential step is someone writing a note "Good for 10 cigarettes". On an individual scale this will also happen quickly. The note may not be recognized as being worth anything, but the honour of the person offering the note is. This will happen as soon as a trader is out of cigarettes and wants to pay his relation. Again, maybe weeks before you see this behavior popping up (we probably need the notion of 'cigarettes as payment' to be established first, because if I offer you useless widgets you won't accept my promise of future payment).
Now the difficult phase: adopting these paper notes (or any other 'good for' token) on a larger scale.
Look at how this has happened historically: anyone with substantial power and (exclusive) access to X can enforce the use of X as a payment system: If you don't take my note/coins as payment, I will chop your head off, because you are questioning my honor. Besides, you can't buy anything in my stores with cigarettes, only with X.
Note that in your clone scenario there will be people/companies with power from the start. Not that the US government, Microsoft or Unilever will chop your head off ;-)
I think we have money now.
So the short answer to your question title is: within a period of months or years we will have at least large regional currencies, but given our worldwide trades between powerful factions I would not be surprised if we have a new global currency in one month.
Your final questions:
Would they, technologically seen, get behind on the first world? 'Economically' may be a more relevant term. Hardly. Trade will slow down for the period we need to establish the new currencies.
And would the gap between "wealthy" and "poor" get higher? Probably, given that existing powerful entities play a large role in this process, I assume they will get richer.
Note that all these developments are just conversations. If you get general agreement that X is worth anything, X is essentially money. (Just look at all the Alternative currencies that exists today).