Here's an update for those that are interested.
Fedex quoted me £202.18 for International Priority or £123.28 for International Economy, and then in the next sentence said they couldn't send the item because it contained hazardous batteries and the destination city wouldn't take it. Makes me wonder why they bothered quoting.
ParcelForce said they could take it, but would have to confirm all the measurements and contents. They said they couldn't quote me before doing that.
Tuffnells (who delivered it initially) wouldn't take it as I didn't have an account with them, and was then referred to Alpha Global International who quoted me £195.00 + VAT and would take 5-7 days to deliver, and an additional 2 days to raise a Dangerous Goods Notice. Also, as I wasn't an account holder they would only take payment via bank transfer or a cleared cheque.
UPS said that I would require a pre-approved contract to ship a package with lithium batteries. As I don't have an account with them, I didn't pursue that any further.
TNT Direct initially said that it was a full Dangerous Goods shipment and that I should speak with TNT Special Services. I called them, and after answering some questions was told that because the battery pack was contained within the motor housing (it came under UN3481 - Lithium Ion Batteries Contained in Equipment), I could send it via TNT Express. They also said that as I didn't have an account (or company under which I could send it), that I should use one of the courier aggregation services, but wouldn't recommend one in particular.
DHL and DX Freight never got back to me.
In the end, I sent the parcel via TNT Economy using a courier aggregation service and the cost with text message notification of delivery cost £46.44. Incidentally, anything containing lithium ion batteries is on the prohibited list for most aggregation services including the one I used, but I ignored that as TNT themselves had said it was OK.
TNT picked the package up on Monday afternoon and delivered it this morning. Only 44 hours. I made sure to attach Ronny's form to the package. I also taped a big label to the side of the box stating that TNT Special Services had approved the package for carriage by TNT Express.
In short, it would appear that TNT should be your first port of call when looking for someone to transport a lithium ion battery pack.