Could well be one or more glow plugs have failed, but this is also a symptom of poor fuel atomisation caused by worn injectors. Basically, a diesel squirts it's fuel into the chamber at very very high pressure through a series of tiny but carefully profiles holes in the nozzle. This fuel is effectively atomised into as fine a "mist" as possible, so it can ignite and burn rapidly. Because a diesel engine uses compression pressure to ignite its fuel (no spark plugs) when it is cold, the "end of compression" temperature may not be quite high enough to start the fuel charge burning properly. Old indirectly injected diesels which had much much lower injection pressures used to struggle to start well at low temps often failing to start at all below around -10degc etc, but the modern high pressure ones, like your PD engine, are much better, and when in good condition will start down to around -25degC without assistance (and on "winter" grade diesel).
But, things like the pump and injectors do wear. They operate under huge loads and typically, after around 100kmiles the injector nozzles will no longer be properly atomising the fuel. When the engine is warm this doesn't matter, as there is plenty of temperature in the "hot" engine to light and burn the fuel. You might get a bit more black smoke sometimes under transients, but most drivers don't notice that. However, when cold, the poorly atomised fuel will be difficult to light, and on starting an entire cylinder or more may fail to fire, and that unburnt fuel will be ejected out into the exhaust, where it will burn very slowly and at low temp, causing a lot of blue/grey smoke to be created (interestingly, the red arrows use a diesel injection system into the tail pipe of their jet engined Hawk aircraft to make their smoke trails so typical of their displays!)
Unfortunately, to "fix" this issue requires replacement (or reconditioning) of your injectors, which is an expensive proposition (typically £100 per injector without at labour costs).
So, get the glow plugs checked, if they are easy to remove this is very easy to do (google it), but in some engines they have been known to seize into the head causing issues (same with injectors). If these are fine, and no smoke is generated when doing a warm start or when driving the car normally, then i'm afraid it's "new" injector time....
BTW. no damage is being caused to the car when it runs rough and smokes like that at start, it just looks crap to your neighbours!