I think Tim W has approached it from the point of view that A is the mortgagor (person who owns the property over which the mortgage is registered) owing money to an unnamed mortgagee (person who holds the mortgage registered over the mortgagor's property). I'm assuming instead that A is the mortgagee holding security over an unnamed other person's property.
I'm also assuming this is not a consumer credit regulated mortgage, which would add a whole bunch of other complexity.
In that case that A is the mortgagee, and its not consumer credit:
- Most mortgages have terms written in allowing a mortgagee to assign their interest without consent. If that's not there, the mortgagor's consent is needed for there to be a valid assignment.
- B would have an equitable interest in the mortgage upon purchase, but not a registered one.
- B could not enforce the mortgage against the mortgagor without notice according to the rules of equity.
- B could not enforce the mortgage against the mortgagor without becoming the registered mortgagee anyway under most (all?) states' titling laws. As far as I am aware, all states and territories in Australia use a Torrens titling system which requires registration to grant indefeasibility (priority of interest). No registration = no indefeasibility.
It's still a valid purchase, it's just only enforceable against A in equity/contract.