Just scoured the area (needle, haystack!!). Unfortunately no sign, if people local to that patch could look at night or early morning, may be lucky, if indeed the poor little soul is still in the area. Fingers crossed.
Just to say it is definitely not safe to leave your dog tied up for any length of time outside a shop. The charities that deal with lost and stolen dogs are advising to never leave your dog unattended. Dogs are getting stolen even from locked cars. It is sad that life has come to this but for the sake of the dog and the ensuing grief from this, please be aware.
Sorry you have lost your dog, yes put his details on the lost and found on here and every other site you can. If you are on Facebook, there is a page dedicated to lost dachshunds. Hope he will be home soon. Did his tag have your phone number on? I assume you have phoned Battersea.
Yes, this person's picture has got to be publicized asap. Have you managed to get posters up in the area? Unfortunately I don't have a printer but am happy to put some up if you can supply. PM me if there is anything I can do to help. I will walk that area today on my dog walk. Thinking of you and your dear little dog.
Have you asked Sainsburys to check their cctv and, if you haven't get his/her details on to DogLost and as many sites as possible. Is he/she microchipped and neutered or spayed. Don't forget to get a crime reference number from the Police. All us animal lovers will do all we can to help, I am so sorry.