The shifting on Alivio is not as good as it could be, I am suspicious that Shimano do this deliberately so that there is a reason to upgrade to the Deore components.
You can put an XT derailleur on there, that will give improved shifting response and work with your existing components. Try changing the cables first as this is more likely to improve the shifting.
The front derailleur can be similarly upgraded.
To get nine-speed on the back will cost lots of money because the cassette is highly engineered (and not just bits of stamped out steel).
If you get your existing Alivio to work correctly by putting new cables in, fixing that front chain-ring and giving everything a good service you will be spending modest sums of money and getting good results. As for spending a small fortune on an XT groupset a piece at a time, I doubt you will be able to go that much quicker on it or the bike be that much lighter.
if you fix what you have and save the money that you would spend on XT then you will have a good deposit for your 'dream bike'. Couple this with a bike-to-work scheme and you might be able to actually afford it, with payments for the rest spread out over the year.
To take a car analogy, you could upgrade a VW Beetle to a Porsche 911 a piece at a time, starting with the transmission. However, this is a lot of work and the Beetle will never really be a Porsche. You would do better to replace with VW Beetle spares and save up for the Porsche instead.
Aftermarket parts always cost more than they should - add up the RRP for all the parts on your bike and it will cost twice what you paid for it. Going for the deluxe parts makes no sense with your situation - although Wiggle/Shimano would like you to believe otherwise.