My understanding, but no expert on this...
The neighbour/adjoining owner (AO) should respond within 14 days. The response should be a consent or a dispute.
If they dispute they should then either appoint their own surveyor, or appoint a joint/agreed surveyor within 10 days.
In practice people usuaully dispute and appoint a surveyor at the same time as many people will get an surveyor to look at the proposals at time of serving notice. (I think scrutinising the notice would even be deemed chargeable so withdrawing the notice after a dispute has probably already incurred some time/fees under the act).
If they AO appoints their "own surveyor" the person servicing notice is liable for the fees still.
If the AO does not respond within 14 days of the initial notice then the dispute is deemed to exist, at that point you are then to tell them they have 10 days to appoint a surveyor or you will appoint one on their behalf. So basically in practice it actually gives an AO 24 days to dispute and appoint a surveyor assuming you give the second notice immediately after 14 days (if you say gave the notice after 15 days they would have 10 days from then) - it seems they have acted in time (they probably don't understand this if they are backdating letters but it doesn't really help your case).
The assumed dispute mechanism exists to stop an AO from refusing to respond and holding things up indefinitely. It doesn't stop things from happening sooner to my understanding.
The notice exists until you withdraw it. And, as is often the case, an AO's surveyor responds for the AO and notifies you they are acting, and by this point they have probably already clocked up a few hours work in scrutinising the notice and information served and sending letters.
In summary - in this instance the main questions are probably:
1. when did you withdraw the notice?
- whilst not done formally and in the wording of the act, your probably could argue this was day 16/17 after the AO has told you they have disputed and appointed their own surveyor and you told them you no longer intend to do works covered under the act.
2. what work if any the AO's surveyor may or may not have done within the limitations of the act before you withdrew it
- maybe a limited amount of reviewing of notices and correspondence
There is quite a bit or mess in all this, but once you told them you no longer inted to carry out works falling under the act I think it would be fair to say the AO surveyor should have stopped working immediately. Its also not your obligation to pay fees/time for proving you later works no longer require an award. However, some initial time/costs may be liable as outlined above.