You should be fine with using the Asda SIM card in the Vodafone phone. Although it is never guaranteed but you've got a pretty good chance.
The main two ways that phones are locked are
1) Network locked
2) Service provider locked
If the phone is Network Locked, it will work with both Vodafone and any provider that uses the Vodafone network.
If the phone is Service Provider locked, it will work with that service provider, but not any other (even if they run on the same network).
Most phones if they are locked, are network locked. Except mobiles on the Virtual networks e.g Asda, Virgin, TalkTalk etc which are usually Service Provider locked - the reason why you can't use the parent network SIM with the phone.