No navigation menu increases conversion rate during checkout process
In the checkout process having a navigation menu can lead to users clicking on it, thus leaving the checkout process. If you remove the menu options users have a lower chance to get distracted, and abandon the process. This has been show in several case studies, see here, here and here. According to them, you should expect around 100% increase in your conversion rate.
Whether to use an alert
If the user have put a lot of effort and want to quit, it'll be a good idea to inform them whether their data is saved and ask them if they are sure they want to exit. However, if you plan to place a nav menu and trigger alert when users are clicking on it, I think it will be better to just remove/hide the navigation.
Ideally, you would present users with minimally viable information during the process keeping users attention on the task. Anything unrelated to it has the possibility to distract them, thus leading to abandonment of the process.