Yes Google home will play Apple Music, but it really wasn't designed for Apple. So it's a bit kludgy. You must Connect your phone using Bluetooth to your Google home and then stream through your phone.
You can play music that's stored on your mobile device or stream from iCloud on Google Home using Bluetooth. Here are the detailed steps (from the web).
Step 1: Pair your mobile device & Google Home
From Google Home app: Open the Google Home app, in the top right corner of the app Home screen, tap Devices
. Find the device card for the Google Home you want to pair with Bluetooth. In the top right corner of the device card, tap the device card menu
> Settings > Paired Bluetooth devices > Enable Pairing Mode.
From your mobile device's Bluetooth settings: On your mobile device's Home screen, tap Settings and then turn on Bluetooth. Then tap the name of the Google Home you'd like to pair.
Step 2: Connect your mobile device & Google Home
Open Bluetooth settings on your mobile device. Under “Available devices”, tap the Google Home you paired to your mobile device.
Step 3: Play your music
After you've paired your phone with the Google Home, it will work like any other Bluetooth speaker. You can play audio from unsupported music streaming services, like Apple Music.
BTW, the Apple Home Pod is amazing.... save up for it! The sound quality will blow you away!