This appears to be a bug with 7.15.0 of Babel which they are making a E2E React native regression test for.
Updated answer
A few hours ago they have released the fix. It is due to a change to @babel/plugin-transform-react-display-name which has been reverted in 7.15.1. With that fix, you should be able to remove the extra entries in the package.json from my original answer and instead do the following.
npm install --save-dev @babel/core@latest
Original answer
Unfortunately, simply downgrading @babel/core to 7.14 will not be enough because it seems to pull the latest dependencies for the other components. Instead what you need to do is lock down @babel/core and any other Babel component you are working that is coercing @babel/types to go to 7.15.0
Here's the relevant section I had in my package.json
"@babel/core": "~7.14.0",
"@babel/compat-data": "~7.14.0",
"@babel/generator": "~7.14.0",
"@babel/helper-compilation-targets": "~7.14.0",
"@babel/helper-create-class-features-plugin": "~7.14.0",
"@babel/helper-create-regexp-features-plugin": "~7.14.0",
"@babel/helper-member-expression-to-functions": "~7.14.0",
"@babel/helper-module-transforms": "~7.14.0",
"@babel/helper-replace-supers": "~7.14.0",
"@babel/plugin-transform-react-display-name": "~7.14.0",
"@babel/plugin-transform-runtime": "~7.14.0",
"@babel/types": "~7.14.0",
Just replace all of that and perform an npm install and let it try to resolve and test your scenario again. If it still fails just look at package-lock.json and locate 7.15.0 and adjust the dependency accordingly
With this approach you don't need to "recover" from a zip file.