BEF mobilisation options WW1

How much leeway does Britain have with the mobilisation and transport to France of the BEF? IOTL 5 divisions were in place in France for the battle of Mons on 23 Aug, while the 4th division began embarking for France on the same day, I believe it didn't actually reach the front lines in France until 31 Aug. The 6th division, the final pre-war complete division in the UK, embarked for France on 8 Sep, joining III Corps which was formed with 4th division on 31 Aug. IIUC about the only other regular unit available in the UK was the Royal Marine Brigade which went to Zebreggue on 27 Aug and withdrew 31 Aug.

Then there is a big time gap until the 7th division, made up of garrison battalions from the closer outposts, and the Royal Naval Division are sent into action in Belgium on 6 Oct.

How much leeway does Britain have in this chain of events?
  • Could they declare war and mobilise a few days later and still have 4 infantry divisions in place at Mons on 23 Aug?
  • Could they have all 6 infantry divisions in the UK at Mons by 23 Aug with OTL DoW/Mobilisation?
  • Could a delayed DoW/Mobilisation have all 6 infantry divisions at Mons by 23 Aug?
 
IIRC, Barbara Tuchman in The Guns of August says that the original BEF deployment plan was for all 6 divisions to go to France immediately, but the UK deviated from the plan to retain the 4th and 6th divisions for training newly raised troops and sent them along later as you mention in the OP. I believe that Christopher Clark corroborates that information in The Sleepwalkers, but I could be misremembering. In any case, the easiest POD seems to be having Britain follow the original plan and send all 6 divisions to the continent immediately.
 
I think the 6 divisions could have all been moved at same time, rather than staggered. How late could Britain leave the DoW to get the BEF to Mons, even if only 4 divisions?
 
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