I am afraid I cannot help with the phone number but live flight information is available via the internet:

http://www.gatwickairport.com/portal/site/default/menuitem.c98f226e4e5cdafe78a5e...

1 - If you work for a firm, then surely it is their responsibility to have a screen on in the office, monitoring flights. Just what are you paying your circuit fees for?
2 - Unless you are sat on the airport, then you won't be doing too many runs a day, and the fee per run should more than cover the 30p max that you would be spending.
3 - If you were parked in the car-park, instead of the McDonalds car-park (don't forget you only get 40 mins there) then you could walk across the gangway into the arrivals hall and watch the monitors. Especially if you "meet & greet"
4 - If for whatever reason you are spending more than 75p per flight, then sign up for the BAA Messenger service, where they will text you the details, up to  and including baggage in the hall.

Don't get upset. I do this for a living as well. I figure that if I get a coffee in the McD's or the BP, then I can use their parking. Good value for 69p. Time it that you arrive a half hour after the flight has landed. European flights are out 1hr after landing. American about 1hr20.

Orange charges 10p per min for an 0870.
The key sequence is 1, 1, 1 + flight no.
About 30 secs. Total 5p

You could use 18185, at 3p per min via their landline access number, but then you would be using inclusive mins, + 3p per min, + 4p connection.
Stick with ringing it direct.
Like several other numbers, this is an information line, supplied for your benefit.
Of course, don't forget that you can claim your phone bill off your tax. So you're not really paying for it.

Before asking for a request, it is a good idea to look on the database first to see if the number you are looking for is there. In your case it is 

Have you tried the unverified number... 01293 503333 ?

But if the automated service kicks in on the geographical number, what is the problem?

The automated message only kicks in when you dial the 0870 number but not on any other number I can find in the database.  I am trying to find the automated message via the 01293 number.  As for using the internet, that is all well and good but I need up to date arrival times and I have to check these after I've left home, so it has to be done by phone.