There is also the possibility that they are telling the truth. Put yourself in their position for a minute. This has happened to me, and it turned out that the courier had dumped the parcel in a neighbour's garden, behind their garage. It only turned up about 3 weeks later when he was gardening. They claimed to have left a card as well, and they hadn't. Keep pursuing the courier until they can prove delivery to an address. Someone signing "Fred Bloggs" on a delivery note is useless.
So far the courier has provided you with nothing. The ball is in their court for now. I notice you make an issue of the delay in advising you. The goods were due for delivery within 3 days of the 20th. Therefore delivery could have been made up to 25th (allowing for the weekend) and then you would give another day until at least 26th before being concerned. So the 29th is not an excessive delay.