When you press the start button the first time (without pressing the clutch), think of it like turning the key to the first click, ie it only puts on the radio etc. When you press it a second time (still without pressing the clutch), think of it like turning the key to the second click, ie all the dash lights come on, including the glow plug light. This is when the glow plugs start to heat up. The start button light won't be green at this point, since you haven't fulfilled the criteria for the engine to start, ie you haven't pressed the clutch. The glow plug light should go out after a few seconds, telling you that it's 'ready', and when you press the clutch the start button will turn green, allowing you to start the car. It should start fairly quickly and without serious rough running, ie it should start 'clean'.
Although the glow plug light should go out after a few seconds, the glow plugs will continue to heat for a period (usually about a minute) even if you haven't started the engine, and they also continue to heat the engine for a few seconds after the engine has started (called 'afterglow').
If yours isn't starting 'clean' it would suggest that one (or more) of your glow plugs isn't working.
Depending on how competent you are under the bonnet there's a fairly easy way to test them. Take the electric lead off the top of the glow plug. Take a jump lead or similar and connect it to the poz battery terminal. Spark the other end of the jump lead against the tip of the glow plug. If you get a big juicy spark then the plug is good, if you get a weak woosy spark or no spark at all then the plug is goosed. If you can't get the end of your jump lead in to the glow plug area then hold something like a long screwdriver against the glow plug connection (thus extending the power feed to your screwdriver shaft) and spark your jump lead against the screwdriver shaft. Sometimes all the glow plugs connections are connected by a single large power bar, if yours has this then you need to remove it first. And if you're not competent under the bonnet then don't try any of this at all, since sparks under the bonnet can cause a fire.
Assuming you find a goosed glow plug then they swap out like spark plugs, but they're usually tight - very tight. Use a decent quality well fitting long reach hex socket (not a multi point socket) on a good tommy bar, and do it with a hot engine (and beware not to scald yourself). I usually replace all four while I'm in there, since they're all probably the same age and if one has gone the others probably aren't far behind.
Alternatively your local mechanic should be able to fit a new set for about £100.
Hope this helps :)