In the book I mentioned (1968?) there is no 15: 16 is Waterloo
to Effingham Jct or Guildford via Epsom.

15 had two uses in the period 1961-81: Soton Terminus - Salisbury via
Nursling, and Waterloo to Alton via Chertsey. The 15 prior to this
was, as the OP said, a short 16. After this, as I was refering to it,
it became a long 17.

I see that it still is, with 42 for via Cobham. In the book it gives
42 as Waterloo - Effingham Jct or Guildford via Brentford Central
- surely that must be a mistake, as I can't see how you can get
from Brentford to Effingham Jct without reversing either at
Twickenham and New Malden, or Surbiton, or Guildford.

1939:
42 Guildford via Cobham
43 Effingham Junction via Cobham

1961:
42 Effingham Junction/Guildford via Cobham (and also Reading - Bognor
via Chertsey, Worplesdon, and Havant; and Portsmouth Harbour -
Southampton Central stopping via Netley)
(43 now Portsmouth & Southsea - Southampton Central stopping via
Netley)

1981:
42 Effingham Junction/Guildford via Cobham
(43 now abolished)

42 is currently one of the more confusing headcodes. Some 42s cross
onto the Fast lines at Surbiton, others stay on the Slow. Perhaps 43
should be reused for the Fast services.