London Proof stamps are the BNP et al stamps. Required by English law on any milsurp sold through there. Nothing to do with London though. The 'BNP' stands for Birmingham Nitro Proof.
"...It was checked by a gunsmith..." He check the headspace with proper gauges? Not many smithy's have .303 British headspace gauges. Not even up here. Good headspace will save you a pile of money. No. 1 rifle bolt heads, that you need a handful of to check with gauges and are not marked by length, run about $15 each.
Anyway, it's a No. 1 Mk III in what appears to be at least Very Good condition. Hard to tell condition from pictures. Restoring a milsurp to military is getting expensive. Stocks and the metal bits for them are getting scarce and pricy. There is no point spending the money on a rifle that is not going to give you a decent restored rifle. Gunparts, for example, wants $5.05 for a rear sight protector(wings around the rear sight). $2.55 for the screw that holds it on. Fore stocks run about $60. Adds up.
Assuming the headspace is ok, measure the barrel length. Should be 25" or very close to that. If that's ok, check the bore condition and slug the barrel. That'd be hammering a cast .30 calibre bullet or suitably sized lead fishing sinker though the barrel from the chamber end using a 1/4" brass rod and a plastic mallet, then measuring it with a micrometer. Lee-Enfield barrels can measure from .311" to .315" and still be considered ok. Over .315" is shot out. However, current factory ammo and reloading bullets come in .311" or .312" so over .312" really is not good. Cast bullets can be loaded though.
