At 13,000 miles, Its really time for a look at the cam chain. Sorry to be the bearer of such good news. As for it dying once its warmed up, you sure youve got the idle set right? Needs to be turning 1,000 rpm or a hair better. Low idle starves the top end for oil, wrecking the head bearings that the cam spins in,. So, just cuz it sounds cool with a low idle isnt a good enough reason to let it idle slow.

Hi again
thanks for the reply, the engine idles at a bit over 1000 revs, this was checked after the first incident by my friend, who is very mechanical minded and fixes all the bikes for the local bike club, but he has never worked on a savage before.

I have a Clymers manual that I got with the bike, the previous owner was also a bike enthusiast who unfortunately had to give up biking due to illness, he hadn't used it for a year or so but had kept it mechanically sound and thankfully knew about the cam chain issue and had it replaced at the last  service and MOT.

I'm not sure how to explain all of this to the mechanic down the road and am a little wary of doing so without knowing the terminology etc, being a woman I always feel patronised by mechanics and would like at least to seem to know a little about the subject before approaching them for help, so any info here would be appreciated.

The friend who looked at the bike last year said it might need gaskets changed because it is burning oil, but to be honest he could have told me it needed a new engine and a flux capacitor and I would have believed him!  possibly not the flux capacitor I might  not have fallen for that one, but only because I have seen back to the future more than once .  
I'm not totally hopeless honest! I can change oil and filters, check tyre pressure etc but when people start talking stripping engines I do get worried

1st off... welcome  

you can get to know your bike better by reading the owners manual.
if you don't have one... here's a copy...
http://suzukisavage.com/cgi-bin/YaBB.pl?board=tech;action=display;num=1138554254
there are also links to the repair manual, but one in hand is always best.

just to be sure, the prime position is with the lever pulled back.

cutting out after a few minutes usually is a petcock problem as ours is vacuum operated.  and if the vac line goes bad, it will shut off.

another possibility is the fuel level is low, so you could need to be on reserve (that's with the lever pushed forward)

another cause is the screen on the inlet of the petcock is clogged or varnished over.  To check that you'll need to drain the fuel and remove the petcock.

now as to "burning oil" to be technical, there's burning oil and leaking oil.  burning oil a gasket kit won't fix.  leaking oil can be, but in many cases, tightening a bolt or an application of RTV will do.  Let us know where it leaks and we can point you in the right direction.

There are two safety cut-outs,.. one for the sidestand and one for the clutch lever...  if these connectors get corroded, they will cause an intermittent stall as you describe...
Disconnect,... clean with wd-40, contact cleaner, or rubbing alcohol,... and reconnect these plugs... at the left handlebar lever (two wire).. and under the seat (it goes down to the sidestand)..

Also,.. here's a complete test for the petcock... it's an easy test, and a bad petcock will cause some crazy symptoms... Always rule it out as a possibility before seeking other remedies...
Test your Petcock,.. don't just turn it to Prime..

Best of luck and welcome... ...

The carb could be running rich as well.
Runs ok when cold (like when it needs the choke on) but as it warms up and needs the leaner mix as when the choke is off the "too rich" mixture causes it to falter and die.
When your friend cleaned the carb did he take note of what size jets were in the carb ?   Or if the white spacer was still on the jet needle ?

One thing for sure, it doesn't really matter how well your Savage is running w/ keys, if you don't have the keys

But while you don't have keys, why not remove the fuel line from the carburator, then put the petcock in "prime" position, (lever toward the back) and see that there is a steady flow of petrol from the hose.
If there is, you have eliminated any problem of running out of fuel when you get on the road.
If there isn't, you must find the restriction by removing the petcock/screen assy.