You're right that the problem is cultural at its root -- there's no one solution to "making" programmers eat their vegetables. But it helps to have many automated reminders, so that the nagging doesn't cause people to hate a single member of the team: when a reviewer is added or a review is updated, your tool should remind the reviewer(s) on a regular basis that new work is pending.
I've been working on a better review tool for teams who have built their process around GitHub pull requests (https://www.omniref.com/code_review), and one thing I've learned from talking with teams is that GitHub's notification system is just comically broken for a team review workflow -- completely overwhelming with emails, but in the wrong way (i.e. a notification for every edit). People just ignore it. So perhaps you should check out a different tool?
> GitHub's notification system is just comically broken for a team review workflow -- completely overwhelming with emails, but in the wrong way (i.e. a notification for every edit). People just ignore it.
Yup. I've found it to be pretty useless for exactly that reason.
Omniref looks awesome. In my experience one of the biggest bottlenecks for getting things done is understanding code bases - wether it's new hires, working with a new repo at your company, or digging into an open source library you use to figure out a bug. I often wish I had better context for why things were done the way they were. Integrating PR comments, and other annotations, into the code, in a way that's easy to browse, sounds incredibly helpful. I will definitely check it out.
I've been working on a better review tool for teams who have built their process around GitHub pull requests (https://www.omniref.com/code_review), and one thing I've learned from talking with teams is that GitHub's notification system is just comically broken for a team review workflow -- completely overwhelming with emails, but in the wrong way (i.e. a notification for every edit). People just ignore it. So perhaps you should check out a different tool?