What would the Catholic Church's hierarchy look like without the Vatican?

Even though the OP states, '[t]his is ecclesiastical theoretical contemplation,' it is not far-fetched; it is a serious question that I believe Catholics ought to consider.

Would the Church continue?
Yes, that's the promise of Christ.

In what form?
This is where this answer aligns with Geremia's answer by looking at history.

What has been is what will be, and what has been done is what will be done; and there is nothing new under the sun. - Qo 1:9 (RSVCE).

Geremia's answer has done a good job presenting past sede vacante periods.

I believe the question wants to find out how bad it can get and to that I look to Judaism.

Will it be as what Judaism looks now without the Temple and its priests?
I do not know, but I believe this is the worst it can get.

We know that the Church has to pass through an ultimate trial before the Lord returns and perhaps in this trial is the desolating sacrifice.

Please see these Catholic End times prophecies.

The Church can exist without a pope. It exists during interregnums after a pope dies and before a new pope is elected; these interregnums can and have lasted years.

Catholicism would cease if the institution of the papacy were destroyed; assassinating a pope does not destroy the papacy. The papacy can in no way be destroyed because that would mean the "gates of hell" (i.e., the heretics) would have prevailed over the Church, which is against Jesus's promise in Matt. 16:18.

When there is no pope, "Who would be in charge?" I think what you really want to know is what would happen if a papal election did not occur or if vying factions organized multiple papal elections (this has happened; cf. the Great Western Schism, for example).

Here's what the prominant ecclesiologist A. Dorsch says (Insitutiones Theologiae Fundamentalis. Innsbruck: Rauch 1928., vol. 2 de Ecclesia, pp. 196-7):

The Church therefore is a society that is essentially monarchical. But this does not prevent the Church, for a short time after the death of a pope, or even for many years, from remaining deprived of her head. Her monarchical form also remains intact in this state…

Thus the Church is then indeed a headless body… Her monarchical form of government remains, though then in a different way — that is, it remains incomplete and to be completed. The ordering of the whole to submission to her Primate is present, even though actual submission is not...

For this reason, the See of Rome is rightly said to remain after the person sitting in it has died — for the See of Rome consists essentially in the rights of the Primate. These rights are an essential and necessary element of the Church. With them, moreover, the Primacy then continues, at least morally. The perennial physical presence of the person of the head, however, is not so strictly necessary.

Most of the cardinals don't live (most of the time) in the neighborhood of the Vatican. So, probably, once the Church as a whole got over the shock, they'd convene the college of cardinals in some appropriate place and elect a new Pope.

Now if (as Caleb hypothesizes in his comment) the college of cardinals were largely or entirely wiped out as well, then we'd probably need to appoint new cardinals. Typically, of course, this is reserved for the Pope; but in this case we'd obviously have to work in a new way. One might consider the Code of Canon Law:

Can. [i.e. Canon] 335. When the Roman See is vacant or entirely impeded, nothing is to be altered in the governance of the universal Church; the special laws issued for these circumstances, however, are to be observed.
Can. 336 The college of bishops, whose head is the Supreme Pontiff and whose members are bishops by virtue of sacramental consecration and hierarchical communion with the head and members of the college and in which the apostolic body continues, together with its head and never without this head, is also the subject of supreme and full power offer [sic, i.e. "over"] the universal Church.

Now it's arguable whether this is enough to allow the college of bishops to appoint cardinals if the see is vacant and there are none; but I'm guessing that it probably is.

It's highly unlikely that all the cardinals would be in Rome at once; I'm not sure that they always are even for a conclave. If there are any cardinals at all left, the college of bishops might only have to indicate that the restriction on voting for cardinals over the age of 80 has been lifted; then a new conclave can be called. If there are no cardinals left at all, then as I said I think you could use Canons 335 and 336, or something similar, to argue that the college of bishops must in these extraordinary circumstances receive the ability sede vacante to appoint new cardinals. From there the process would proceed as usual.

A council of bishops might be needed to determine where the Holy See would be, but the new Pope would be able to appoint new heads of the various congregations and the other structures of the Holy See. It would certainly be a great blow to the Church, but the Church has a widely dispersed power structure and would be able to pick itself up fairly easily, I think.

One other note: I'm not familiar with the governance of the Eastern Catholic Churches, nor with the canon law regarding them, so I've left them out; but presumably their patriarchs would not have been affected by a catastrophe such as you imagine. Unless you want to specify that there were quite a number of simultaneous catastrophes in different places.