Don't worry about the Dutch side, they have very good internet at low prices. Be sure that on the UK side the you have the highest upload available. (Go for 1Mb or higher) Slingmedia says that 256 kb is enough, but that's the bigges lie ever. 512 kb will do the job, but on a TV screen the image looks poor (ok for a small window on your pc)

But why using a slingbox ? Amsterdam is in the footprint of the Astra satelites, so all you have to do is to subscribe to UK satelite TV, bring your setup box and smartcard with you, and install a small dish in your Amsterdam house.

For legal reasons, UK subscription packages are only sold in the UK, so yes, you will have to carry your current setup to the Netherlands. Just check the local authorities and the landlord if there are any restrictions for the dish.