So, since Git uses SHA-1, does this mean we're going to see a new major version number of Git that uses SHA-2 or SHA-3 in a few years?

I don't expect one overnight. For one, as noted, this is a collision attack, one which took a large scale of power to achieve. In light of that, I don't think the integrity of git repos is in immediate danger. So I don't think it'd be an immediate concern of the the Git devs.

Secondly, wouldn't moving to SHA-2 or SHA-3 be a compatibility-breaking change? I'd think that would be painful to deal with, especially the larger the code base, or the more activity it sees. Linux itself would be a worst-case scenario in that regard. But, it can be pulled off for Linux, then I'd think any other code base should be achievable.