Hmm-m, I don't recall anyone claiming <= 15 minutes, though it's not that
far off thanks to its small saturated boiler. This is what I reported in the La Marzocco Linea Mini Review
Well I have claimed it
Probably also here in some old post and recently on a Dutch coffee forum. But... first of all mine is also 230V.
I've measured with a stopwatch. After about 8 minutes the heating light and the machine can be activated. After about 12 minutes you can hear that the heating in the brew group switches off: relays clicks are audible and typical water heat sound of heating elements stops... Maybe thats different in the 110v version if less powerful heating elements are installed but in the 230v version you can hear it.
It will make espresso at this point, I've tried.
However the temperature will not yet be stable when doing anything else than a small cup. This is because the steam boiler keeps heating up until about 18 min. You can monitor this on the pressure gauge.
This will influence doing longer shots since the Mini works a bit differently than most machines. It takes pre heated water from a sizeable hx in the steam boiler and mixes this with cold reservoir water. This provides feed water within a certain temperature range which is then fed to the brew group which does the final bit of heating via pid control. It's been a while but his has been elaborated upon somewhere in the loooong thread LM user experience thread quite a while ago by someone who studied a disassembled machine. So basically you can look at it as a hybrid between a hx and a double boiler machine. It explains the quick warm up and rapid recovery between shots.
After 18 minutes, on a 230v model, parts of the housing will still heat up further. I guess this takes about 30-40 minutes. The brew path is stabilized quite quickly though.
P.S., I leave it on 24/7 as well most of the time. Same reasons as stated in this thread earlier.