What is the correct way to say "It was this week that Justin and my lives changed forever"?
The least amount of restructuring I can think of is:
It was this week that both my life and Justin's changed forever.
The use of both makes it clear that there are two lives—rather than a shared single life (in the case of a partnership).
Without any restructuring, the "best" that can be done (I think) is:
It was this week that Justin's and my life changed forever.
With joint ownership, the possessive belongs to the final subject. In the example, since the first subject has a possessive, it could be assumed that there is no joint ownership. Of course, this is still not ideal. I don't think there is a good solution without some degree of restructuring.