Is there a way to include a link to update the payment details in the email notifications (V1)?

Yes, there is. Just go to Settings >> Customer-facing essentials >> Email Notifications >> hit Customize Template corresponding to the appropriate email notification.

Copy the customer.payment_method_update_url mail merge field in the email template so that customers receive a URL to update the payment method in the email. The mail merge fields are available below the email template. Screenshot is below.

Note: The mail merge fields are not uniform across all the email notifications. For a particular email, only the mail merge fields applicable to that event will be available.

For example, the payment method update URL merge field is applicable to a payment failed email as the customer has to update the payment method so that the next attempt is successful.

If you are using the V2 email notifications, you can check this article.