Bike Case on British Rail

I'm considering traveling by rail between London-Gatwick and Cambridge
next month and I'm wondering what grief I might expect with a bike
case in tow. What's the transfer like between the London Kings Cross
station and the Kings Cross Thameslink station? What about a cab in
Cambridge?

Here's the bike case I'm considering:

http://tinyurl.com/2647ut

Weight: 26 lbs.
Dimensions: 45" x 34 x 11.5"

Thanks.

--
Daniel