If you're just talking about "rotating" the track so it becomes Edinburgh-London-Edinburgh, that should be fairly straightforward. Many of the online mapping sites (e.g. RideWithGPS is my default recommendation) will let you upload a big GPX, cut it into pieces, and save it back out. There are also tools to stitch GPX tracks together. (In a pinch, some copy/paste work using Notepad will work - GPX is a simple text format.)
In the particular case of LEL, it's even easier, because the route is published as several separate GPX files, one for each leg between controls. You could ride them in whatever order you choose.
Making more detailed changes to a GPX track is not necessarily easy, because a GPX track doesn't retain information on where the control points (in the general sense of "go here then here then here", not the Audax sense) along the route are. If the route was made using an online mapping site you've got a good chance of being able to make your own copy and editing it using that site's tools, though.