Well, I don't think for a minute that's it's real. The big giveaway for me is that the letter is clearly written with an entirely different pen to the signature.
The letter has a thicker grey ballpoint ink, the signature is with fine pen that's dark black and very thin.
Even if the signature was real, what often happens is people get an autograph at a concert with a blank piece of paper (whatever they could find at the moment) and then the text could be added to it afterwards. so there may be absolutely no connect between the top and the bottom.
The only way to give some vague credence to this is to trawl old archive images to see if he was ever wearing such a watch, if he wore a watch at all.
To add - the watch looks very well worn at the rear, as in worn out. So logic tells you he would have worn it a lot.
But trawling through a few hundred images on Google, I have yet to see a shot of him wearing a watch at all, period. That in my mind pretty much says it all. (athough I read he had Patek, gifted from Yoko)
This seller cant be serious can he?
A quick google search reveals a lot of confirmed hand written letters from John Lennon.
When comparing this its clearly fake.
"30-Jan-18 at 07:48:16 AEDST, seller added the following information:
Hi everyone
Just to remind you all that if auction price is below $10,000 the listing will be cancelled, this is a very precious item to my family and especially to my grandfather so please be aware of this because if it doesn’t reach past this limit I will cancel auction and please don’t send me questions saying why did you cancel bla bla. Thanks for being aware of this all the best and GOOD LUCK!!!"