There’s 2 (well, 3) types of weight restriction. Environmental and structural (and axle) limits.
Environmental limits are signed by the picture of the truck with the with weight limit in the picture, e.g. 7.5T. These are usually put up to stop trucks using residential/unsuitable roads as rat-runs, generally as a response to nimbyism. You can get a fine, but it’s a non-endorseable offence, and normally included in areas where they have decriminalised enforcement (i.e., it’s a council fine). Plenty of “trading standards sent me a fine” stuff going around, dunno whether that’s true or not. Often you’ll have except for access, or except for loading/unloading exemptions on a plate below the sign.
Structural weight limits are signed simply with the weight limit with MGW below it. So, if a sign says e.g. 18t MGW, it means no vehicles over 18t MGW - i.e., whats on your ministry plate. Some limits are as low as 3t! The sign will normally be accompanied by a ‘Weak Road’ or ‘Weak Bridge’ plate below it. These are put in place where the road/bridge is/may structurally be unable to handle anything much heaver than the signed weight, or if repeatedly sending heavier vehicles over it could cause damage.
The only exemption to this is where it’s plated below it ‘Except Unladen Vehicles’, where you can go over it if your MGW is higher than the limit, and, IIRC, even if your unladen weight is higher than the limit.
Going over these is endorsable, 3 points + £100 fine, I believe.
Axle limits are the same as the above MGW limits, normally these signs appear on longer bridges/stretches of road that can’t cope with the weight imposed by a single axle. These seem to be much rarer than the normal MGW limits, but I’d take them more seriously, it seems that they are signed this way because imposing X weight from a single axle is likely to cause the bloody thing to collapse. AFAIK, it’s again the plated axle-weight, regardless of being loaded or not (unless signed otherwise).
Supposedly the reason behind it being a MGW limit rather than an ‘actual weight at the time’ limit, is to make enforecement easier. They don’t need to drag you to a weighbridge, they just check your ministry plate!
HTH 
As an aside, on one of our runs, most of the drivers use a particular route which includes going over a narrow railway over-bridge with an 18T MGW limit. The trucks can weigh anything up to 32T when they are going over (yes, nearly double the limit). I refuse to use it and happily take the 10-15min detour… I keep telling these muppets one day they’ll come across a copper having a bad day, and they’ll find themselves with a fine and license endorsement. Oh well. More worrying though is that it’s a bridge over a high-speed railway line. Imagine you went over it way over weight, part of it collapsed onto the line and a train hit it. Consequences don’t bear thinking about, but not one of these useless ■■■■■ gives a toss. Maybe they think their foreign license will save them from a manslaugter charge too? 
completely my own fault, no complaints.
… and then WE have to apply for an exemption!