Transportation from London to Haworth in December

That's a long trip when the days are short.

Be careful if you are planning on traveling on Christmas Eve, Christmas Day or Boxing Day, as transportation will be very thin on the ground or non-existent.

Train from Kings Cross to Leeds, change for the Northern Rail service from Leeds to Morecambe or Skipton and alight at Keighley. Onwards by bus or taxi, about 10 miles. http://www.keighleybus.co.uk/

Bus choices on the Keighley and District page are:-

500 Keighley - Haworth - Hebden Bridge via Ingrow, Cross Roads and Oxenhope
664 Keighley - Haworth - Stanbury via Ingrow and Cross Roads
665 Keighley - Haworth - Oakworth via Ingrow and Cross Roads
720 Keighley - Oxenhope & Leeming via Spring Bank, Ingrow, Bracken Bank, Oakworth, Haworth and Haworth Brow

and once you are there, if it runs in the winter -
812 BRONTË SCENIC TOUR: Haworth - Scar Top Pine
via Brontë Village, Brontë Parsonage, Stanbury, Oxenhope

There's also the Keighley and Worth Valley Railway between Keighley and Haworth. It's more a visitor attraction than a means of transport, but it's one of the top preserved lines in Britain and definitely worth visiting if it's running while you are there. http://kwvr.co.uk/ During December they will be running the usual "Santa Specials" which are very popular and may need advanced reservations.

"Also would appreciate tips on what the weather would be like in both London and Haworth."

Cold and damp probably, maybe a bit of snow ooooop north (Haworth) but our weather is unpredictable, so nothing is certain.

Mahalo for all of your fast replies. I am very grateful for all the information. If the trip is too long should I consider staying overnight somewhere? Any suggestions for a nice stay close to a train station between London and Haworth?

No point in staying somewhere on the way. London to Leeds is two to two-and-a-half hours.