Your plan sounds fine.
If it were me, I would likely do the temporary support from beneath (a temporary stud extending from the floor below, intersecting the ceiling at the intersection of B and P). Either way, I think you will be fine.
A reciprocating saw is the right tool for this job. It will allow you to cut the nails holding the existing cross member in place.
I'm not sure of your particular situation, but since you are already reframing the opening, you should consider adding a drop-down ladder. As long as you have the space, this will make the attic much more usable in general.
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.
Best Answer
For occassional storage chipboard or any plywood 1/2" or greater will be fine. Go to 5/8 minimum if your joists are on 2' centers.
I personally don't agree with the advice to use plywood. Chipboard is fine: it's strong, and it's made using waste materials so it's more ecologically sound to use it where it's appropriate. It's not appropriate for applications that are exposed to the elements in any way: in those cases plywood is more durable so justifies the 2x cost and the extra environmental impact.
Chipboard does have a strength axis, which you might as well take advantage of it: you want the joists running across the short side, i.e. 5 joists on every 4x8 sheet if the joists are on 2' centers.
Chipboard made for flooring has a tongue-and-groove along the long side. That's because while the short sides are supported by the joists, the long sides bridge the joists. Locking a tongue into a groove prevents sagging between the floor joists. Obviously not a big deal for you storage loft, but if it only costs a few more bucks for the tongue-and-groove I'd get that. I've only seen that in 3/4" and 1", BTW, so if you get that you'll have a proper floor up there that you could hold a neighborhood dance on.