Q. Hard lines are the walls on the first floor underneath, joists on the first floor landing run left to right so does that mean the central wall running front to back is 'load bearing'?
If your house is in the UK (which I'm guessing it is), then without a doubt the central wall running from front to back is load-bearing.
Q. Initially I thought of putting 8x2s front to back, supporting them on internal walls where possible. Problem is that the angle of the eaves would not allow me to rest the ends on anything substantial so could I just bolt them to the rafters as long as they were supported by the walls downstairs too?
Refer to this question here on SE, and read all the answers and comments found within.
Running parallel to the existing ceiling joists is the easiest and preferred way to go.
Removing the binders "locally" to allow the new floor joists to pass is "normally" considered acceptable (then connecting/pinning them back to the new joists passing through), though you would want to check such a detail with your local Building Authority before doing so.
Otherwise if you go the other way (perpendicular), you end up going above the existing ceiling joists, which causes a major re-work of the existing wall plate area that sits on the exterior wall and the roof area above.
Q. Then I got thinking that I'd need to go left to right .. a) because the load bearing walls go down the centre of the house and b) because if I don't the ceiling binder will be in the way. Same problem however with regards to getting the joists to sit on the external walls and I'm guessing I'm not allowed to bolt a plate to the chimney wall and hang things off that.
Whatever you do, do not connect anything to the chimney wall. Doing so is an absolute no, no in the UK (if I have your location correct).
Again, refer to this question here on SE, and read all the answers and comments found within.
Of course another option is to remove the existing ceilings (lath and plaster), binders, etc. Leave the existing ceiling joists and install new appropriately sized floor/ceiling joists parallel to the existing, then fix a new plasterboard ceiling to the new joists. Granted, such an option adds considerable mess to the job, and a reasonable cost increase.
Look where the joists meet the ridge beam where you're going to put the door.
Get some dimensional lumber about 4 feet long, at least as thick as the existing beam. Nailing a couple of 2x6's together would probably suffice in the short term.
Give yourself enough room to work, and using a jack post and this 4 foot beam segment, support the joists in the area where you are going to work. Don't over support, as you'll lift the floor above. The idea is that you want to take the weight off the wall where you're working. Jack the post until it's snug, and take out one wall stud in the middle of the door. Check the supported joists for level, and jack up if necessary.
If required, remove another stud, and adjust the jack again.
Put in your door frame and header, and slowly jack the temporary beam down.
Best Answer
As @TheEvilGreebo mentions, 2x6 is not enough for a floor that will be used as "habitable space". I know because when I remodeled my attic to convert it to a bedroom, the existing 2x6 floor joists (over a very similar span) were not enough and I had to sister them all.
At a minimum you will need to check with your local (municipal or county) building department. They should be able to tell you what you need.
It might be easier to get a structural engineer or architect to come out and tell you what to do. If they are not drawing up plans you might be surprised that their prices are quite reasonable (they should be able to give you a quote for what it would cost for them to come out and generate a report up front). It will also help when you get your building permit to be able to submit the report written by a certified professional to the building department.