If you can take them with you great but if you can't then just make them as safe as you can in your home. A tight squeeze for a little while probably won't be a big deal but I am not there and don't know the situation so use your judgement.
If you do use a travel cage then use common sense. They will have to be loaded in the travel cage well before you go to the shelter as in whenever there is a tornado warning in your immediate area because tornadoes aren't forgiving they don't wait for you to grab animals.
Please remember people are most important if you have to make a split second decision you choose yourself and human family and hope for the best with your animals. Ideally that won't happen but I can tell you from experience animals sense bad weather and panic. They will not always cooperate so it is best to have them settled where they need to be far in advance.
I survived F4 tornado that took out half my town. I have been in trenches dealt with panicked animals and made split second decisions and all this with the luxury of having a shelter inside our home I don't want to imagine how much more hectic it will be running across a yard with a tornado in the vicinity.
This is a photo of my old house taken 36 hours after the tornado. That field behind the house isn't a field. It was a neighborhood that was filled with fences, trees and houses just 2 days before.
Yes, try to save your birds if you can but don't wait because anything in the path of a tornado will be destroyed.
If you do use a travel cage then use common sense. They will have to be loaded in the travel cage well before you go to the shelter as in whenever there is a tornado warning in your immediate area because tornadoes aren't forgiving they don't wait for you to grab animals.
Please remember people are most important if you have to make a split second decision you choose yourself and human family and hope for the best with your animals. Ideally that won't happen but I can tell you from experience animals sense bad weather and panic. They will not always cooperate so it is best to have them settled where they need to be far in advance.
I survived F4 tornado that took out half my town. I have been in trenches dealt with panicked animals and made split second decisions and all this with the luxury of having a shelter inside our home I don't want to imagine how much more hectic it will be running across a yard with a tornado in the vicinity.
This is a photo of my old house taken 36 hours after the tornado. That field behind the house isn't a field. It was a neighborhood that was filled with fences, trees and houses just 2 days before.

Yes, try to save your birds if you can but don't wait because anything in the path of a tornado will be destroyed.