Wow all these answers to the fixing problem, i am impressed.
A couple of things.
1/. i [wrongly] assumed that you had entire walls of either breeze or Plaster. your reply, implies that indeed the walls are pilastered, but there is a small area of Breeze below the plaster line? If so be careful that this "gap" is not large enough to if when hit by the toe of a shoe, or the vacuum cleaner the skirting you are about to fix buckles back and into the gap. if the gap that is showing Breeze is more than about one third of the overall height of the new skirtings then the gap needs to be filled, this to stop the skirting being displaced as above, also, if you screw into a relatively "Un-Supported" area of the skirting that is not in actual contact with the plaster, the gap where you see the Breeze will allow the skirting to be deformed grossly out of shape. Glad that bit is over, now for the confusing bit?
2/. How to get the skirting in behind the Rad, given it can not be slid down from the top
A/. possibly most "elegant" solution, cut a piece of skirting to the exact size between the center lines of the Rads pipes, place in position, fix and leave.
B/. Cut the other two bits of skirting from corner to the fixed bit of skirting at the Rad, the pipes will in effect "hide" the joint
3/. Depending on the dimensions of the room, you can "feed" a length of the new skirting in behind the rad pipes and get a joint somewhere in the middle / or near the opposing rad pipe, fix and leave, the rest of the cut to the opposing corner can be fed in from the other side of the radiator.
Drawback of this solution is how do you get the glue on to the back of the first bit you feed in? bend it and hope that you obtain a good enough area of glue to hold it on.
Best bet is option one. the joints ar "masked" by the rad pipes themselves?
Hope all this is making some sort of sense?
You also mention that you are going to screw and plug the skirting? be cautious, you will need a fairly robust [thick] skirting to allow you to do this, the more you remove the less strength left in the skirting, more prone to splitting.