ET3D: I think it depends a lot on Williams, but I'll come to it a bit later.

Being on opposite sides isn't by itself a huge barrier. It's still a matter of trust. People are taught that their enemies are out to get them, but anyone who isn't a fanatic will likely be willing to accept a person of the other side as a person once it's clear that the "enemy" doesn't wish him or her harm.

In this case both are already outcasts from their own groups of their own will, and don't believe that things are black and white. Both have reasons to stick together, Williams for finding what happened to her family, the mage for protection from his or her own group. So this partly falls back to the question of whether they could have been friends had they been in the same group.

I assume that you want them to be friends, and I think, given the above, that it's entirely plausible (if we assume that only the circumstances are keeping them from being friends). Still, a lot of it falls to how Williams will behave. Williams has already saved the mage, so the mage would think of her favourably. On the other hand, Williams only wants to use the mage, and if she reveals that tactlessly, the mage is likely to feel she's no different than the illegal mages.

Here it's up to you to determine how their behaviour will help or hinder the friendship. Williams will likely need to be convinced that the mage is more than just a tool. This could come from finding something they share (like loss of their families) or from the mage doing something "beyond the call of duty", such as saving her life back in the extreme case, or just showing gratitude in nice ways. The mage may be a little paranoid at this point, but if Williams treats him or her right that might alleviate the fears.

tl;dr: These are two people who have good reasons to work together; they aren't at cross purposes. The obstacle to their friendship is realising that.

KneeTheCap: Thank you for this, it was a great read and has good points!