Is it possible to georeference using just UK postal codes. I have the minimum free account and have had now luck with this. IF so should the postal codes be continuous strings with no gaps and is it correct to add in the free text box the country as "United Kingdom"?
Yes it is, simply follows the steps in this tutorial and you should get what you need. I tested it on a table I have stored using this postcode column and the co-ordinates were swiftly generated.
As you write in your question, I added 'United Kingdom' in the country field and it worked.