I think I've got him.
Living in Reading would point towards one of the two Berkshire RHA batteries. Pre-war there was one but became two at the outbreak of war, 1/1st and 2/1st. A territorial artilleryman, as with all other territorials, started with one number and received a new one in 1917 when the TF were renumbered. The Berkshire batteries received new numbers between 616000 and 61800 so we are looking for a man with two numbers, the second in that bracket. 1/1st Berkshire Battery went to Egypt in early 1915 and remained there.
644, later 616243 Driver Fred G Carter RHA fits well. Where he lived and his second number point to the Berkshire Battery, he went to Egypt on 25 April 1915. In his picture he is dressed as a driver.
Two other men going to Egypt at the same time (I didn't look for more) were 640 later 616240 and 641 later 616241. The second of these is in hospital in Alexandria with malaria in 1919 identified as Berkshire Battery. 632 later 616234 was a Reading man. I could cross check others but frankly those are enough for me.
All that said, only his medal card and associated medal award roll survive (you probably know 60% or so of service records were lost in WW2 to bombing).
This is what Longlongtrail has on the Berkshire Battery.
Battery HQ was at Reading with a Section at Ascot, and the unit was allocated as artillery support to the 2nd South Midland Mounted Brigade
The battery was duplicated in 1914 forming 1/1st and 2/1st Berkshire Batteries.
1/1st served in the UK until 1915 when it went to the Middle East for the rest of the war where it initially served as artillery support to the 22nd Mounted Brigade. In 1917 it moved to the 6th Mounted Brigade, and later the same year formed 20th Brigade RHA TF with the Hampshire and Leicestershire Batteries.MaxD
Lots on the net:
https://en.wikipedia.org/wiki/Berkshire_Royal_Horse_Artillery#1/1st_Berkshire including some images.