I have had a look in the books and found a bit of info on this. Rather than shifts, horses were "resting" - though what this exactly entails is still not clear to me.
According to Tony Atkins "GWR Goods Cartage Vol 1", a GWR report of 1869 found that the GWR had an average of 7½% of its horse stock "resting". This was critiqued as being lower than other companies (a private cartage agency is quoted as having 29% resting at this time).
In Janet Russel's "GWR Horsepower" these figures are given for selected London stables in 1877:
And here again at a much later date when the GWR had started doing more cartage of its own, and greatly increased the number of horses. No date but from the wording of the text it sounds like the 1920s (edit: just found the same numbers quoted in Atkinson, he says the following is "just before WW1").
There is a now a "sick" category, but the number of resting horses is quite low.
Returning to Chris' question about number of horsedrawn vehicles, I have oddly not been able so far to find any data on this for particular goods yards. However, as the years pass by Farthing is slowly growing in my mind, from a medium sized junction station to a fairly large one. So I hope there will be room for a few more horsedrawn vehicles. If not, I'll have to invent some sort of special industry as an excuse - such as the Witney blanket industry:
http://witneyblanketstory.org.uk/WBP.asp?navigationPage=Transport
https://www.steampicturelibrary.com/stations-halts/london-stations-paddington-station/paddington-goods-depot-6300310.html