how long does it take to add my card to my wallet

If you applied through the Wallet then it should be there once approved. Are you talking about a card other than Apple Card? It should only take seconds

1. Sometimes if you sign out of your Apple ID and back in it resolves the problem.
2. Make sure you are using the latest iOS/macOS update.
3. Check your Apple ID region to make sure it is correct. If it is correct, you may need to select the same region again. (Change your Apple ID country or region - Apple Support)
4. Check the region on your device to ensure it is correct (Settings>general>language & region - set to your region)
5. A passcode is required to use the Wallet app. 
6. You need to be in a region that supports Apple Pay - Countries and regions that support Apple Pay - Apple Support
7. Your bank needs to support Apple Pay - Apple Pay participating banks - Apple Support
8. If none of these suggestions are able to resolve your issue you can go here for further assistance - If you can't add cards to Wallet for Apple Pay - Apple Support
9. Apple Pay support can be reached here - Apple Pay Customers

• If you need to change your default payment in Wallet or have other issues such as the inability to add passes you can go to these links which covers managing Wallet and wallet use.
• Add and manage passes in Wallet on iPhone - Apple Support, please note that passes cannot be directly added within wallet, except for QR code enabled passes. For everything else, its done either through the associated app, an email, airdrop or a link in a website.
• If you get an error that you already have the maximum amount of cards in wallet with less than 22 cards signing out of your Apple ID and back in usually resolves this issue.
• If you are getting a message to contact the card issuer, call the bank or credit card company who issued you the card to make sure there are no restrictions on the card. 
• If you are getting a message that Apple Pay services are unavailable at this time, sign out of your Apple ID and back in.
• How to use Wallet on your iPhone, iPod touch, and Apple Watch - Apple Support,
• Manage the cards that you use with Apple Pay - Apple Support

Typically the process takes a few minutes. You will need to enter the requested data. Depending on the card provider you may be instructed to go through a few extra steps to validate that the card is yours. Once that is done, it should only take seconds for your card to be usable on Wallet.

You should sign out of your Apple ID and back in.