thenateo, welcome to FT.
The first Heathrow Express actually doesn't leave until 5.10am, arriving at T1/2/3 at 5.26am. You should probably allow 10 minutes to reach the check-in desks from there, although it should be a little less.
Whilst there are no outgoing passport checks from the UK, I would still be nervous about only arriving at the terminal with less than an hour to spare - it's just enough time, particularly with the Star Alliance Fast Track security (where you may need to manually show your Gold card - the machines will not accept a non-J boarding pass, even though the Heathrow Airport staff always make you try) - but not really sufficient to be comfortable. If anything goes wrong, you'll be more than sweating.
Fortunately, LH usually leaves from T5A where their lounge is, so at least you're spared the 10 minutes across to B terminal.
I would personally take the 4.42 train. Note that it's a Heathrow Connect, not a Heathrow Express, so Express tickets are not valid (and vice versa). Allow some time in the morning to purchase a ticket, which you can do at either the machines (the regular ones, not the ones on the Express platforms, and make sure you buy for the Connect), or at a ticket office. (Alternatively, you could buy in advance online and collect at a machine, which would be a bit quicker.) The trains generally leave from the platforms around the corner from the main bank - and it will take two or three minutes to get to them - so if you're unfamiliar with the station layout I'd arrive there no later than 4.20am.
Last edited by NWIFlyer; Jul 12, 2016 at 7:35 am