Being the mom of a 4 yr. old and 1 yr. old I of course defer to all the moms that will be posting here who have actually pulled off transitions with older children.
That said, I think that gradual changes can be your best tool - that and gluten-free and/or grain-free adaptations of neolithic favorites.
Example: Breaded, fried items like chicken nuggets can be fairly easily adapted and even prepared perhaps slightly faster than the time to bake traditional chicken nuggets in the oven - dip thin strips of raw chicken breast in an egg wash (1 or 2 eggs, raw, scrambled, with salt and pepper), and then in a mix of 1 part coconut flour, 1 part unsweetened shredded coconut (add other seasonings if you like). Fry in coconut oil. This breading combo also works nicely on raw peeled shrimp and thin fillets of fish like turbot (flounder) or tilapia.
Paleo baked goods could be your friend in a slow transition - and especially because they often include pumpkin, sweet potato, and/or pureed fruits that your kids might otherwise not be willing to consume. I realize that you are working under unique circumstances with your sons' preferences, too, so you have the knowledge of what types of baked goods they'd be most likely to try initially. There are so many great blogs with baked good offerings, but for starters I suggest checking out the grain-free baked good recipes over at Joyful Abode and also Life As A Plate. My family has recently tried a grain-free mix for Chebe pizza crust - which is a great way to sneak in some cheese - maybe Kerrygold Dubliner or Ballyshannon? - with great success. ETA: The Chebe crust instructions allow the maker to mix up to 1 cup of shredded cheese into the crust besides the cheese you'd use to top, so it could be a standalone cheese integrated thing if melted cheese as a topping is not an option.
Good luck! Turning the ship around (as it were) is so hard - and you have even more challenging circumstances. I would be so curious to know how you and your family are faring in the coming weeks and months.