Probably not a week since there are other parasitic loads besides the lamps, but the current drain is definintely MUCH less into the boot light which was the thing that originally triggered the whole 'project' off.
In reality there's a lot more to it because the battery self-discharges somewhat over time anyway so that compounds anything else that's going on, and depending on the condition of the starter motor, solenoid, etc. even with a good battery, on a really cold day when the chemistry in the battery is 'slowed down' due to the drop in temp it can still take a fair bit to crank over an engine enough to get it to fire and run.
The other thing about colder weather and is that the starter solenoid can take a bit more magnetic field strength before it'll engage the power feed to the main starter windings.
It'd be different for every case, but the overall benefits of going with LED lamps far outweigh any downside as far as my experience is concerned, but they're not a panacea for overcoming other issues with a car's electrical system.
I run with LED lights in the front and rear clusters on both of my registered Saab's. The only lights not LED-based are the H4 headlamp bulbs, but technology will improve to make replacements for those possible in time.
Craig.