In reply to Toby Armstrong:
Conville Course as recommended would be great in a year or two. Meanwhile, your profile suggests you are in a perfect spot to develop some alpine skills and you already have lots of skills ( climbing HVS multipitch trad at Bosi which is very similar to Chamonix granite). Try to avoid the temptation to buy lots of gear just yet!
So, some suggestions:
1) Climb lots of multipitch routes in a range of grades within a day. Focus on moving efficiently, efficient ropework techniques and building stamina.
2) Purchase Bruce Goodlad's book on alpine climbing ( others are available but I think his is especially well written ) and try the relevant techniques at home . For example; ropework for moving together , ropework for glacier travel, ropework for crevasse self rescue and rescuing a partner. Better still, get a friend / volunteer / members of a local club / like minded partner to practice these outside with. Be cautious however; granite is not kind on ropes, practice on very easy terrain first.
3) If you can, as others have suggested, try to get to some mountains for multi day walking, scrambling and climbing linking together routes. North Wales, Lakes and Scotland would be ideal.
4) If you have the opportunity try to get some winter walking / scrambling / climbing in.
5) At the moment your alpine dream may seem a long way away but the alps provide a whole host of different climbing opportunities. With your current skills set and equipment you could go rock climbing in a non glacial alpine environment and get to know areas before taking on fully alpine climbs.