You might have the scars to prove it, but any number of tall stories and battle scars won't convince the sceptical that you've just walked across the weather map. Luckily there is a way to prove that you've walked from one end of the country to the other, as long as you get your act together at the start of your walk.
Back when I did the walk, the Land's End John o'Groats Club (which no longer exists) issued certificates to those who could prove that they had indeed made it across Britain, and they provided me with a verification form before I set off. The form consisted of a blank square for each day of the walk, which I got stamped at Post Offices, signed by B&B hosts or scribbed on by campsite owners at the end of each day. It was actually rather fun getting the form signed, and I looked forward to it each day.
When I finished the walk, I handed in the form along with a small fee, and I was issued with an official certificate confirming that, yes, I was quite bonkers. I also got a year's membership for my trouble, so I could share my experience with other LEJOG aficionados.
I don't know if anyone else provides this service now that the club is no more, but you could do worse than asking at the Land's End John o'Groats Association (they also have a Facebook page). If that seems a bit too much trouble, the Scottish Youth Hostel at John o'Groats is only too happy to issue its own certificate. For this you just have to look like you've come all the way from Land's End; the SYHA are a trusting bunch, which is no bad thing. This certificate is also free, which is an added bonus.
Here are my various verification forms and certificates, just for completeness. They might not look like much, but to me, they're like old friends...