Not a huge number, given the timeline: war was declared at 7pm GMT, effective 11pm. Lord Kitchener was appointed Secretary of State for War on 5 August, and began accelerating armed forces recruitment, but his first major appeal for volunteers wasn't until 7 August.

You have to bear in mind that a 4-year global war of attrition was not widely anticipated — so, in the confused early days of the war, there was a surge in enlistment but not on the massive scale that would come later.

I can't find daily breakdowns of enlistment figures, but a total of 478,893 men joined the army between 4 August and 12 September. There are also some interesting stats on the pace of recruitment in the early weeks/months of the war here.

Recruitment spiked after the Battle of Mons and leading up to the Battle of the Marne, as it became clear that the relatively small British Expeditionary Force (around 80,000 men) would need to be massively reinforced.