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Visit this page on your phone
Click the download button to get the APK. Once downloaded, click to install. You may need to go into your Security Settings to allow Unknown Sources. If so, revisit your download folder to attempt the APK install a second time.
A bit tedious. I see the ALDI TALK app on the Play Store and have never had a phone that was compatible. :( (Labeled incompatible with two differnet unlocked Nexus devices)
As you have done the registration in person, you no longer need that app. You should install the regular ALDI Talk app (https://play.google.com/store/apps/details?id=de.eplus.mappecc.client.android.alditalk) though, so you can book data plans and control the credit balance.
Head over to https://www.alditalk-kundenbetreuung.de and create your account there if you haven't done so. Most services there are the same as in the phone app but there is more, it's also useful to have in case your phone gets lost.
Important: Go to https://www.alditalk-kundenbetreuung.de/de/konto/drittanbieter and make sure that you do a Vollsperre (set it to "yes"). This is to ensure that you don't get robbed by accidentally clicking on something like a fake app button or browser pop-up which triggers an invisible subscription for non-existing phone services at 5 or 10 Euros per month!
Should you ever need to pay anything else via ALDI Talk, you can log in there again and un-block the service you need. In general, most people don't need this, so Vollsperre is the best option.
The convenient option for 30 day plans is to give ALDI your bank account info. But you may want to have full control and just top up the SIM with vouchers. You can get them in every ALDI, starting from 5 Euros but you can also purchase any E-Plus voucher in other supermarkets or online at amazon.de (gift cards). They work because ALDI Talk used to be E-Plus (which now is O2 including E-Plus).
Feel free to ask if you have more questions :)