Truth be told, I think that it's possible that the HR people at 102 New Street aren't all reading off the same song sheet. The firm answers to all the questions I have had have come direct from the lady that handled this recruitment drive, Sheila Spence. I think all the time I've rung LM since they offered me a job on the 6th, I've spoken to her, and she's been pretty cool, knows when I need to hand my notice in to my current employer, yadda-yadda-yadda.
The date that I heard bantered around the last time I spoke to LM, was indeed at the beginning of January. It may be that those of you that have been told otherwise have either a) been given misinformation by someone that isn't really too sure, or b) won't be starting until february because they've filled the january intake up.
If you've been offered a job but haven't had a formal offer yet, it's probably because not everyone who is starting the same date as you has had their medical yet. The date for mine had to be changed (at my request, not theirs) as I work perma-nights and have 2 children which made it awkward for me. I'm not now having my medical until next Friday, (14th) so I assume that I'll be one of the last of the current batch to go through it. I would imagine that when this batch are all done, then the offer letters will come out.
As I see it, if they offer you a job and you can't work your notice period with your employer but you don't want to turn the LM job down, then go cap-in-hand to your boss like I did with mine. They can only say no, you can't cut your notice short, so then you leave anyway. They've still go to pay you what you're owed, so bugger them!!!