You did not mention the unshipping problem before, so I am with Bill - check the outer limit adjustment, as the most logical explanation is that you caused a new problem during your adjustment. However, it makes no sense that the chain would come off when shifting to an intermidiate cog, rather than the smallest, unless you are referring to it coming off in the front, rather than ther rear.

As for other causes, we don't know how worn the chain is, as you have not indicated you checked it for wear per the procedure I posted above, and side wear in an older chain can produce shifting problems. We also cannot tell remotely if there are unusual problems with the rear derailleur. So It's not possible to tell which of those could be a problem. In addition if it is unshipping in the front there is a new set of possibiliites. Not knowing the wear on the chain also means it is unknown if you will find problems meshing with the rear cogs if you change the chain. The cable is not causing unshipping but could cause inconsistent shifting if not lubricated properly or binding for some other reason. As I agree the labor prices are high (£60 or $92 per hour!) and it's a long drive to boot this is what I suggest:

1. Measure the chain for wear as I described earlier and refer to the link to Sheldon's site for interpreting the results.
2. Lubricate the cables where they pass through housing. Easiest way is to shift to the largest rear cog and then, without pedaling, shift the lever back to the opposite end, which will leave the cable very slack. At that point you should be able to remove the housing by using a slot that is in the downtube or chainstay housing stop - will look similar to the pic below. Once you have done that you will be able slide the housings enough on the cables to access and inspect the portion hidden by the cables. If there is any rust, broken cable strands or kinked cable/housing replace the cables and housing, preferably with stainless cables. You will need to Google replace bicycle shift cable and study several videos and tutorials so that you understand the entire procedure. If the cables look good, just feed some oil onto each section as you slide the housing back over it and the end back into the stop. Once you are done and the housing is in place turn the cranks and shift back and forth to make sure everything is seated. Always lube non-stainless cables, using a medium heavy oil - not grease.
3. Once you have addressed the chain and cables, recheck the limit adjustments and tension. If you are still having problems let us know everything you did and remaining symptoms.