Yes, they are quite entitled to reject the refund request for now and to try and get the item delivered. You need to be patient. In a couple of weeks, when either the second item will have arrived or not, you can either return the item within 7 days of receipt for a refund (under the distance selling regulations) or ask them to try and get the item delivered or ask for a refund.
It is up to them if they want to keep trying to get the item delivered and, if they keep failing, you can ask them to use a tracked or a different delivery service. Obviously they can't keep doing this for ever but you can only insist on a refund either within 7 days of receipt or when it is obvious that they are not going to manage to deliver the item.
The company's name is The Online Shopping Company Limited and their phone number is 01702 614 430. The person who answers the phone may not have your order records to hand.