Try House Elf http://houseelf.co.uk/about-us/
they do everything and anything and have helped me move twice.
Other moving companies - Aussie Man & Van I have used for moving stuff to a rental out of London as well as moving between storage and rentals.
Both companies really helpful, efficient, professional and fun crowd, and much more reasonably priced than some of the well known big names around here. It might be an idea to move on a Monday or Tuesday rates should be lower and they may have more staff and van availability.