3 hours is a very sensible amount of time for the transfer.
If UK passport is machine readable, then no Visa requirement - entry will be on a green I-94W form. Bags will be collected at TBIT - take the 'A' Bus to T2 for NZ (or walk, 15mins with a cart) - go up one floor from arrivals to departures.
Mexy passport - don't know requirements.