I recently learned that Edom today means the Christians. See for example Ibn Ezra
This is not universally agreed upon. See Shadal on Gen. 27:40 and Ibn Kaspi on Deut. 23:20-21
So given what it says in Ovadiah 1:18... Does this mean we are destined to wipe out all Christians?
It does not apply to ALL individual Edomites as the Gemara (Avodah Zarah 10b) makes clear:
Once he [Antoninus] asked him [Rebbi]: 'Shall I enter the world to
come?' 'Yes!' said Rebbi. 'But,' said Antoninus, 'is it not written,
There will be no remnant to the house of Esau?' 'That,' he replied
'applies only to those whose evil deeds are like to those of Esau.' We
have learnt likewise: There will be no remnant to the House of Esau,
might have been taken to apply to all, therefore Scripture says
distinctly — To the House of Esau, so as to make it apply only to
those who act as Esau did.
So even if the Ibn Ezra and those that identify Christianity with Edom is correct, only those that perpetuate evil Edomite ways will be destroyed (i.e. the "House" but not necessarily individuals).
When will this happen? Before or after Moshiach comes?
The Rambam in H. Melakhim 12 is relevant here:
Some of the Sages say that Elijah will come before the coming of the
Messiah. But regarding all these matters and similar, no one knows how
it will be until it will be. For these matters were unclear to the
Prophets. Even the Sages themselves did not have a Tradition regarding
these matters and only could attempt to understand the verses. Thus,
there were disagreements in these matters. Nevertheless, neither the
order that these events will occur nor their details are fundamental
to the religion. Thus, a person must never busy himself with the
Aggadoth and not dwell on the Midrashim regarding these matters or
similar issues. He must not make them dogma. For these do not result
in either love (for God) nor fear (of sin).Similarly, one should not calculate the Ends. Our Sages have said that the spirit of those who calculate the Ends will expire. Rather, one is to (simply) wait and believe in the principle of this matter, as we have explained.
That said...
Given that the era of Moshiach is characterised by universal peace and disarmament it seems to me that maybe this should happen before Moshiach comes.
The Malbim does suggest that it means that Moshiah ben Yosef will do the job, but again that is speculative and there is not much to be gained from such inquiries on our part.