Private Sector
Provide desktop support to multiple offices (London, South Africa, Netherlands)
First or second line? Locations not too an important factor when thinking about the salary for this job imho.
Administrate and maintain the domain
Do you mean admin within ad users and computers, account admin, or actually managing the domain infrastructure? Not knocking you but from experience (I know as I've done it in the past!) a lot of the time all this means is you add people to groups and reset passwords
See above, do you do the basic mailbox management or actually manage the exchange infrastructure?
Build new servers if needed
This one seems a little odd and doesn't really fit with the rest of the role to me. Do you just whack an image/os on and hand over or do everything?
Travel to offices around the UK (London, Croydon, Southend)
This is another one that wouldn't really matter for the salary imho.
Do you create the backup strategy, troubleshoot, verify/test etc or just change tapes?
I'm not intentionally being an arse if it comes across that way, just it makes a big difference between a 15k job and a 25k job

The guys at my place who do the admin side of all that start on around 20k.
PM's in the Public Sector around here (East Midlands) are on 35-40k. Don't forget the Public Sector pays less (I'm about 10k behind the average for my job but prefer working conditions in the Public Sector).
Personally don't agree that public sector pays less, you can't really say that for all jobs. I'd say I'm paid more in my public sector role that I would be if I was private

That's the reason I changed jobs, they offered me a lot more money
And after all that personally I'd say a firsty line desktop support person who does basic admin and troubleshooting would be on around 16-20k depending on the company, but location plays a huge part.
My wife is first line, it's her first IT job, and whilst not wanting to say exactly what her pay is it's above the range I stated (private co, I think they pay some people there too much and some not enough!).