A 13A Plug Fuse (BS1362) will carry 13A indefinitely, because thats is what it's rated at. Fuses aren't the best device to use for overload protection as they are generally have a wide tolerance. Looking at a Time/Current plot from the Bussmann website, a 13A fuse will take about 1.6 times rated current without blowing, the curve is an exponential one, so it would blow eventually but after many minutes (or longer), however go to 2 times current and the fuse will blow in 0.11Seconds. Into this you will also need to consider, the capability of the plug, its connections and the cable. At 230v 5kW should have given you a current loading of 21.7A by my calculations, is this down to meter accuracy or lack of it? At 230v - a resistive load of 3KW will give slightly over 13A - and you may find the plug gets warm at this! An RCD is protecting against earth leakage - so will only trip when you have an earth fault (a difference between the flow of current down the live and returned via the neutral of more than 30mA), so consequently isn't related to an overload situation.
