At a very high level, the data we need for card activation is stored in the Mastercard Debit application on the card. This is accessible in a similar way to how a payment terminal would interact with the card during a payment. We would need to be able to talk to that application and request the details for activation.
The Core NFC framework on iOS only allows access to NDEF formatted data storage NFC tags and does not allow for raw reads and writes over the NFC interface. 
You can find out a little more about contactless EMV from episode 4 of Monzo Insider. 