Like CX828, I would also be interested to know who is the OTA.
My understanding is that you ask your OTA for a change of date.
If the OTA cancelled your tickets without reissuing new ones, the ball is clearly in their court. Apparently they claim something else.
Are you sure that the original tickets were issued, do you have a ticket number? When you asked the OTA for a change of date, did they advise you that the new dates required repricing to a higher bucket?
In any case, you have to deal with the OTA, not CX. In the past months, CX has seen a huge increase in demand and a small one in capacity. The cheap fare buckets that were available months ago, have now disappeared and unlikely to come back in the near future, including for LHR-HKG. Just get a refund (hopefully without cancellation fee) and try to get the cheapest ticket currently available. Prices keep going up.