I'll vote for a new bike but that isn't a huge surprise

You can change the tires and you will feel a little more efficient once up to speed but by putting on slicks you will not gain in where it matter most for road riding....acceleration and climbing.
I ride a 25 lb mtn bike w/slicks as my commuter and I also have a 20 lb road bike and I can ride them both around the same speeds (advantage road bike though) but the road bike just leaps away from the mtn bike like no tomorrow and when climbing, those 5 lbs make all the difference. Plus, with no shock you are not losing energy with every pedal stroke. Also, the frame will feel more stiff which will transfer more energy into the bike moving forward.
I don't know your budget but this is what we did for my parter after finding that most hybrids still weigh in at/over 25 lbs and have slow wheels.
Bought a Devinci Apex (105-9 speed, R500 wheels) road bike and changed out the drop handlebars for flat bars...presto...a sub 20 lb hybrid bike that is much better than anything we could find. I could see the difference in her riding right away, much better acceleration, easier climbing and less energy used for just riding.
There are still last year road bikes available if you look around. A good shop will change over the bars for zero or little cost.
EDIT: I see you are in London...I have two buddies from there that drove to Toronto to buy a bike and they saved over $500 each. Might be worth the drive.