Deployment to TestFlight is handled by Circle CI. Nightly betas are release automatically, but if you need to deploy something right away, run make deploy locally to trigger a beta build on CI. It takes about 45 minutes. There is a blog post on the process here.
Note that only one beta can be deployed at a time; teams should use feature flags to avoid the need for having two parallel beta versions.
There are two types of betas on TestFlight: Internal and External. Our deploy script sends the beta to both groups. However, Internal testers get access to the beta immediately, while external testers may have a delay of several hours/days while Apple does beta review. This additional review typically only happens when we change the version number.