So eventually got my SSN by going to an SSA office and completing the forms after being here for 3 1/2 weeks and not hearing anything. Very helpful and the document arrived in the post about 10 days later. They said that the process hadn't been started so had to fill out a few forms and show proof of ID, visa and address (thank god I'd opened a bank account already)
Still waiting on the Welcome Notice and the Green Card itself though. For anyone that's already landed and gone through this process, do you know if there's anyway to chase these or find out how long they take to get issued?