I purchased an album as a gift and input the recipient's email address. The purchase went through and was paid for but the recipient never received an email to download the purchased album.

Ask the intended recipient to check:

1.

Check the spam folder of the Web version mailbox.

2.

Check the spam folder of the mail client software.

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See -> Give apps, music, films and more from Apple

Give apps, music, films and more from Apple – Apple Support (UK)

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1.

If the recipient can't find the email with the redeem button, you can resend the gift.

2.

If there's an issue when you send a gift

• The recipient of your gift must have an Apple ID in your country or region. For example, if you're in the United States, you can't send a gift to a friend who lives in Australia.
• In some regions, you can't send gifts. Store availability also varies by country or region. Find out what's available in your country or region.
• If you use Family Sharing and Ask To Buy is turned on for your account, you can't purchase gifts. 
• You can't gift individual songs on an iPhone or iPad. To send a song as a gift, use your Mac or PC.
• You can't send apps, TV programmes or films as a gift on a Mac or PC. To gift those items, use an iPhone or iPad.
• You can't send a free app or item as a gift. But you can tap the Share button  or the arrow button  to share a link to the app or item.

3.

Cancel a gift

If you want to cancel a gift, you can contact Apple Support up to two days before the scheduled delivery date.