After a bit of research, you can go to these 3 places to get an instant travel card:
Post Office (crap rates)
Travelex (so-so rates)
TUI (great rates)
These places will be able to issue you a travel card, then and there. I managed to get Travelex to match TUI's rate - after much haggling. The Travelex was far more convenient to get to for me.
I am waiting for Chase to approve my application - it's quite slow.