I would expect a property developer to maximise his profits ==> do a cheap bodge job.
Old loose render can trap water and cause damp, it is likely to be on random patches on the wall, not particularly near the floor or the ceiling. New render should not trap water at all.
Wet patches starting near the ceiling are probably gutter or roof.
If the extension is fairly modern (50 years or less) it is likely to have cavity walls so wet should not normally track to the inside surface, unless the cavity is running with water and it shows at the bottom of the wall.
If the extension is new it should not have any wet patches at all.
If the vendor was capable of doing it properly, he would have done. He has failed so don't let him have another go. Anyway, you would need to wait three months to see if the walls dried out after repair.
Go round on a rainy day and see if water is running off the roof or the gutters or round the windows. A flat roof should not leak for 20 years or so, but if badly built it might be ponding. Have a look from an upstairs window.
Look up the plans on the council's website and see if it has all been certificated as completed. There is a chance the BCO walked past during the build and leant against it to see if it fell over. BCOs tend to pay more attention to jobs that they are not happy with.