Its been a couple of years since I had my Lelit. The boiler (technically) is heated up within a few minutes, however, you need to give it at least 20 minutes for the whole system (including brew head) to be fully warmed up. On the Lelit, the boiler is situated directly over the group so its a bit quicker than other machines (eg Silvia) and, in my experience, more temperature stable.
One thing that you can do in the mornings is have your machine on a timer switch, set it to switch on about 20 mins or so before you get up (making sure the night before youve filled the tank and dont have the brew, steam or hot water switches on). I still do that with the Giotto, no problems, ready to go when I stumble to the kitchen.
From recollection, after pulling a shot and switching on the steam, it takes about 20 secs or a bit more for the steam to be up to speed. About the time it takes to knock out the puck and pour your milk in the jug. (And dont forget to switch off the steam and re-prime the boiler after steaming.) Of course, if you want to then pour another shot, the boiler takes about 10 mins to come back down to a stable brew temp.
Good machine, the Lelit, mischky. Makes really good coffee, great simple unit.