Put down plastic sheet and cover with old towels.
Use paint roller tray to catch water. Use small jug and bucket to collect water from paint tray.
Fully turn off inlet and outlet (lock shield valve) controls. If the inlet is a TRV turn it to off (0) position of fit the little screw on cap if you have one.
Undo the union and let the water trickle into the paint tray. Undo the bleed screw to let some air in and the water will drain quicker.
Put another paint tray under the other union and undo that. Towards the end the water may come out black.
Close off the bleed screw and lift the rad from the wall - often easier with two people. Turn rad upside down for carrying so black sludge does not drip out on floor.
If you move the brackets the radiator may no longer mate properly with the inlet and outlet pipes so consider another approach to how you solve the problem.
Good luck...
Peccavi, June 2009