Actually we had probs with this as well, (ordered at the end of oct too) though our customer service wasn't as bad as yours.
1st we ordered the dolls house & the furniture, after a couple of days we got an email saying that the house was out of stock & were refunded for it though the order for the furniture was still active.
Ended up checking the Asda site every day till the house came bk into stock & then ordered 1 for home delivery which arrived within a couple of days.
Didn't get a text about the furniture for pickup so phoned customer service after a fortnight or so. Customer service apologised & gave an e-cheque to cover the price of furniture & home delivery, which I then had to wait till it came bk into stock 4 days later to use. Furniture was delivered the next day.
So it was a fair bit of hassle but got the stuff in the end.
Oh, & by the way, we got a text last wed to say that our furniture was at the store for collection but ignored it, so there is prob a bag of furniture sat with our name on it at the local store 
Sorry to hear about all the trouble yo are getting with it, if my experience is anything to go by then I'd say to try complaining to customer service again 