Obviously I don't know timelines but the usual way betas are released for Android if using the Play Store deployment is that you will be sent a link to a Google+ Community or a Google Group (I have a feeling it will be the latter of the 2).
You will then log in with the same Google account you use for the Play Store and will request to join the community/group. I guess Per & the team will then check each request and match it to the responses in the form you completed previously and accept your application. Once you are a mamber of the group there will be instructions to go to a Google Play testing website which allows you to opt in to receiving the beta updates through the standard Google Play Store update delivery method. Again, making sure you are logged into the same account you use for the Play Store click the 'Opt In' button.
Once done it can take up to a few hours (but normally sooner) for your Google Play account to be configured to receive the beta updates and you'll start seeing these updates on your device.
Do be aware that after opting in you may not receive an update to the Spotify app as it can take a while for your account to be configured as well as Google may stage the deployment. So wait until the team confirm they've published a beta release in the group and within a few hours it should show up in the Play Store as an update for you.
Please use the search feature before posting a problem/bug/idea. What's a Spotify Rock Star, and how do I become one? Using Spotify Since March 10th 2009 Twitter: @mechanimal82