It takes 2-3 years for the methane to reach those heights, and 20+ for the carbon dioxide. The average increase in temperature we have now is caused by 2003's emissions.
Other informations about how all this work (basic, incomplete, but better than what I usually hear):
Greenhouse effect doesn't really work like a greenhouse.
Basically, to be in a balanced system, earth has to re-radiate all the energy it gets from the sun. The energy a body radiate depends on its temperature (plank's law). Earth need to reach 255K (-18 basically) to radiate all it gets from the sun. The nitrogen, dioxygen and argon in the atmosphere do not prevent that radiation, but water (especially) and Co2 do.
This basically make that earth cannot radiate the energy it gets at ground level (that's the greenhouse effect) and has to radiate the energy where the atmosphere is thinner, a few kilometers up, where the concentration of both Co2 and h2o is lower (concentration here as particle per m3). So the earth has to heat all the atmosphere until that altitude (where GHGs do not block radiation anymore) reach 255K.
So, when the Co2 concentration above that point rises, the altitude the radiation occurs rises too. But that altitude is colder, it cannot radiate as much energy as it gets from the sun. So the atmosphere heat up a bit, until that new altitude reach 255k.
Obviously it's a lot more complex, with reflection, absorption spectrum of molecules... There is also more than one GHG, there is stratosphere effect, but this explain way better than 'we live in a big greenhouse'.Does that make sense? First time I do this in writing and in English?
I'll take every correction that will not make the explanation too hard (unless I'm flatly wrong and lying instead of vulgarising).