I have been curious about this. There are plenty of casualty figures available, and most armies lost more soldiers during the war than they began with, due to recruiting or drafting more soldiers as the war went on. How likely was a random soldier (who was involved in the war from the very beginning) to survive to the end of it? As an add on, if the data is available, how many of the soldiers deployed in August 1914 also ended up fighting in WW2 and surviving that? Were there any people who fought both wars from start to finish and survived?
I can speak to the first part of your question: how much of the BEF survived until 1918.
The prewar BEF was largely destroyed in 1914. Casualties were just about 90,000 between the start of the campaign in August and 30 November 1914. The original establishment of the first seven infantry divisions was 84,000. By November 1, 75/84 British infantry battalions had fewer than 300 men (they're supposed to have around 1000). The loss of officers was also substantial: 842 killed, 688 missing, and 2097 wounded since August. Reinforcements totaling around 109,000 men in the form of the Indian Corps and Territorial units kept the British Army in the field in time for New Army units to start arriving in 1915.
The problem with these casualty figures is that it's not clear what "casualty" means in this context. By that, I mean we don't know for certain how many of those 90,000 casualties were killed, how many were wounded and taken out of service, or how many were wounded and then reinserted into the line at a later date.
Overall, though, we can say with a good deal of confidence that very few of those who crossed into France with the original BEF fought through until 1918 because the vast majority of that army was destroyed, mostly at Ypres and La Bassee (58,000 casualties between 14 October and 30 November).
Source: These casualty figures are all over the place, but I consulted Beckett, Bowman, and Connelly, The British Army and the First World War (2017), 228-9.
Edit: I can speak a bit more about casualty figures in 1914 in the other Western Front armies, but not the extent or with the same amount of detail that I can about the British Army. The casualties of 1914 were by far the highest of the war in relation to the size of the belligerent armies. French dead alone for 1914 totaled 265,000; by September 10, about one month into the war, the French army suffered 385,000 losses. The Belgians lost half of their army in Flanders in 1914. German losses in Flanders were 80,000, with reserve divisions suffering 60% losses. The ferocity of open and semi-open warfare on the Western Front in 1914 was wholly unsustainable and exhausted all armies of manpower and munitions. These armies, although enormous by 19th century standards, were smaller than what came after in 1916 and 1917, so it's reasonable to assume that a small proportion of the British, French, and German armies in 1918 comprised men who went to war in August 1914. See Strachan, The First World War, volume 1, 278.