I thought "Vectors for the ILS" is different from an SRA? Certainly I'd use a different approach chart anyway. The one at Prestwick has a separate SRA approach chart which includes instructions if you lose radio contact.
During a surveillance radar approach (SRA) the pilot is given distances from touchdown, advisory altitude information and azimuth instructions to enable him to make an approach to a particular runway
That is quite different from "Vectors to the ILS".
Really, an SRA is what happens after you get vectors to the top of the glide slope. I suspect it also requires ATC to have different qualifications. There is also a no-compass / no gyro version, in which they tell you for how long to turn left / right, assuming rate 1 turns.
To answer the original question, I'd keep it simple and say:
G-ABCD, Request SRA
or if feeling extravagant:
G-ABCD, Request surveillance radar approach runway 06
If I wanted a different runway to that on the ATIS, I'd add the runway designator.
BTW - CAP413 is still there - it is just the supplements that have been removed. See page 149 of the pdf in the actual CAP413 (designated Chapter 6 Page 21 if looking in the index).
And should say - vectoring for the ILS is on p137, although it isn't explicit on how the aircraft should request this.