It's difficult to imagine, as replicating the specific situation of the Ashkenazi Jews is tough.
The way it worked for them in OTL was (and I'm simplifying):
1. Jewish groups entered Italy at some point during Classical Antiquity, possibly the late Roman Empire (Rabbinical Judaism forms in about the 300-600 AD, so this sets the upper limits on time depth).
2. As Europe Christianizes, and in the post Roman period, prohibitions of lending to brother adherents in religion become more important. A Jew cannot lend to a Jew with interest, and a Christian, under the law of Western Catholicism, cannot lend to a Christian with interest.
3. During the post Roman early and high Middle Ages we see expansion of urbanisation or civilization (towns, etc.) across the North European plain. (For reasons that are not totally understood) This interacts with 3 to create the problem of a shortage of lenders for Christian townsmen and aristocrats.
4. But! Christianity is *relatively* tolerant of Judaism as a common descendant of the parent Hebrew religion, without any proselytizing character (relative to how tolerant Judaism, Christianity, are to everything else). And Jews are allowed to lend to Christians.
5. So a relatively small group is poised at the right time to enter the business of credit, when no other group is really in a position to do so.
Though the business is quite hard - its involves high risk lending, and Jews were frequently subjects of disappropriation and brutal persecution - this ultimately leads to a quite spectacular era of population growth, as a small community of hundreds of members grows to the millions, in a trajectory unprecedented in Europe across the same time span.
This advantage ultimately recedes by the late Middle Ages and early Modern Age, when in most of Europe changes in religion allow and favour Christian bankers and lenders, though the Ashkenazi persist in numbers in the east.
This isn't the outcome of any endogenous cultural characteristics of Rabbinical Judaism; it's all timing, and the relationship with Catholic Christianity.
The problems for the Roma in replicating this are:
A. They arrive too late. They are unlikely to arrive until the 14th century, which is too late to become established in this role, and even then are not seen in Western Europe until the 15th to 16th centuries, which is far too late.
B. They don't have any special religious exemptions on lending. They generally convert (or are made to convert) to local forms of Christianity, and then the same religious law applies.
C. Finally, even if there was a special exemption for Roma Christians to lend money at interest, its then not likely that they would persist as a minority. The Roma in Eastern Europe show much more intermarriage with Eastern European people, relative to the time of their introduction to Europe.
(Expanding on C: For European Jews it (mostly) seems like the founding population had some European ancestry then very little (almost no) intermarriage during the era of high population growth. While the Roma look like they have continuously intermarried with Southeastern European populations, albeit still at a relatively low rate, and to have "taken on" quite a bit more ancestry from local populations relative to the time they have lived in Europe.
Difference being that the Ashkenazi Jews isolation (less intermarriage) is likely due to religious reasons, which we see in different groups practicing Rabbincal Judaism dispersed across the world, from India to Turkey to North Africa (where they hold no special economic status). The Roma are Christian co-religionists, and the lesser degree of isolation as a group is mostly due to stigmatization.)
With a much more prosperous Roma niche, I would guess this would probably not have happened, and they would cease to be a separate ethnic group.)