Personally, I'm not sure it'd be something that interests a politically united Arabia, or at least internal politics may get in the way.
Say you have a politically united Arabia, but practising up-to 4+ different faiths. Christianity, Zoroastrianism, Judaism, and Pre-Islamic Arabic faiths.
The Christian community may significantly object to invading the Romans, the Zoroastrians the same with the Persians, etc.
So the question is what would bring all four groups together for war, or a majority of them? Trade is making them wealthy, and they can certainly emigrate - and lord knows there is plenty of work as mercenaries for either the Persians or Romans.
More likely, IMO - is they'll wait to be woo'd. From any perspective Syria or Mesopotamia are great conquests. If the Romans are quicker off the mark, they could offer the Arabs significant gold to jointly invade Mesopotamia, giving them the south - and the Romans the North. I personally think this would lead to a long-term political capital in al-Hirah.
Alternatively, the Persians may be quicker off the mark, leading to Syria and Egypt falling, leading to a similar rise for Damascus.
The key here is I'm assuming without Islam you don't have religious unification.
It would be different IMO if they were united - the religion of choice making a difference.
Orthodox Arabs, with the Patriarch of Jerusalem as their Patriarch, may be a strong ally - and supplant Aksum as an ally for the Romans.
If Zoroastrian, I would expect the reverse.
Judaism IMO makes Arabia an ally for the Persians, in order to reclaim Jerusalem, with Egypt a natural following conquest.
I have no knowledge of Arabian Paganism, so... *shrug*.
The issue is it would be hard to get a politically united Arabia given all the tribes, small states, and kingdoms there was at the time. Without Islam, or one big bending religion it would be messy getting it done in uniting Arabia. Early Islam had it civil wars and schism in the form of the Fitna as it was.