I disagree with gehngis... Normally (for what that's worth) I would suggest a MATZ crossing clearance will include clearance through the ATZ (unless specific routing instructions are passed e.g. route no closer that 3nms etc)
In the example you quote, the situation might be more complex... I would hope that Farnboro would liaise with ODI before issuing the clearance... and unless otherwise instructed negate the need to speak to ODI themselves, but that may not be reflected in the letter of agreement between the units... so "if in doubt, ask!"