There won't be any evidence offered, because this event is almost certainly a work of fiction. A careful reading of the articles and simply thinking things through will reveal colossal, gaping holes in the story the British government is peddling.
Firstly: we know beyond doubt that this story is at least partly fictional. We know this because the anonymous government sources (i.e. civil service officials) keep contradicting each other. We see for example this quote in the Independent, "However, despite a senior government official was quoted by the paper as saying that Snowden had "blood on his hands", Downing Street confirmed that there was “no evidence of anyone being harmed” as a result of his leaks". Different versions of the same story contradicting each other is a good sign that what we're being fed is a story: things always grow in the telling, especially when we're hearing a third or fourth hand account of what happened. The way US officials contradicted each other in the wake of the bin Laden assassination is a good example of that.
Secondly: this story asks us believe several extraordinary and completely implausible things.
In the UK foreign spying with people is the mandate of MI6, a separate agency to GCHQ, which handles signals intelligence only. It's like the split between the CIA and the NSA. Yet in several years of Snowden reporting there has never been any mention of documents from MI6. There has in fact only been a single mention of MI6 in the GCHQ/NSA documents, and that was a joint presentation about spying on climate change conferences! So the UK government is asking us to believe that journalists like Greenwald (who hates the UK because of the holding of his partner at Heathrow) would have a large cache of documents from an entirely separate agency and yet find nothing newsworthy in them at all ..... indeed, apparently MI6 is so boring that the existence of such documents isn't even worth mentioning? Apparently the UK has never done anything even embarrassing in many years of engaging in foreign HUMINT? That stretches the bounds of credulity beyond breaking point.
But it goes on. We are asked to swallow a second utterly ridiculous idea. Apparently the Russians and Chinese suddenly got access to a wealth of information on British spies, information so detailed it allowed them to be targeted:
The newspaper quoted a senior Home Office source as saying: “Putin didn't give him asylum for nothing. His documents were encrypted but they weren't completely secure and we have now seen our agents and assets being targeted.”
What normally happens when spies are caught? Well, they are normally arrested and tried, or at minimum thrown out of the country. Yet Downing Street is telling us that there was "no evidence of anyone being harmed". In short, we're being asked to believe that Russian and Chinese counter-intelligence suddenly found themselves with information so detailed that it amounts to a brain-dump of MI6, including lists of foreign agents ...... yet they walked away from the biggest gift in counter-intel history with nothing at all. Not a single arrest, not a single trial.
That the KGB and Chinese counter-intelligence are so incompetent defies belief - indeed, it is literally unbelievable.
There's a third totally implausible thing about this story. It asks us to believe that there is a cache of encrypted Snowden documents out there .... somewhere ..... and the Russians/Chinese were both able to obtain this cache, yet they could not obtain the accompanying password. So where did this cache come from? Again, the civil service is asking us to believe something utterly stupid: "Putin didn't give him asylum for nothing" .... in other words, he was given asylum in return for a secret cache of unreadable documents, that Snowden did not have any ability to unlock. What a great deal for Putin! Such a story makes little sense, and is also contradicted by Snowden himself: he said many times he did not take any documents with him. Once they were given to the journalists, that was it. And he clearly sees himself as an American patriot, so such a move would make sense: he knew he must leave America, and he knew if he had the documents he'd be forced to give them up. Solution: leave without the documents.
So if the story we're being fed here is just propaganda, where did it come from?
I strongly suspect that the genesis of this story is some kind of internal report from MI6 or GCHQ that tries to explain a drop in foreign intel performance. The home office official quoted above says only that "they have been targeting our agents and assets". I suspect the story evolved like this:
1) MI6/GCHQ - the quality of our intel has gone down. The Russians/Chinese seem to be harder to spy on than before. We're moving agents around to try and fix things. Possible explanations include a mole inside our operations, improvements in RU/CN counter-intel, or information gleaned from the Snowden leaks.
2) Home office flunky - the quality of our intel has gone down. The Russians/Chinese are using the Snowden documents to stop us from spying on them. We're being forced to move agents around.
3) More senior home office flunky - we've had an intelligence disaster. The RU/CNs have cracked the encryption on the Snowden documents and that forced us to move our agents. We're lucky no-one was killed. Snowden could have had blood on his hands.
4) Home office "official leaker to the press" - there's been a massive intelligence disaster, Snowden has blood on his hands after our enemies cracked the encryption and started targeting our agents and assets, we have been forced to move spies to keep them out of harms way.
With each iteration up the management chain the story grows in the telling. The "cracked the encryption" part is pure Hollywood - nobody who knows anything about encryption or hacking can really believe this story. But it's the sort of thing that'd sound absolutely convincing to a middle aged civil servants with an arts degree whose entire knowledge of spycraft comes from the movies.
Actually, I'll take my speculation a step further - I strongly suspect that in reality MI6 has very few or no agents inside Russia or China and they were almost entirely reliant on GCHQ SIGINT operations for insight there. Partly because their staff are all so busy running around after jihadi's, partly because the people running those agencies know that Russia and China aren't all that dangerous, and partly because pre-Snowden SIGINT was so much more effective. We know this because of this story from the British press in March which quotes the Foreign Secretary as saying:
Making a speech earlier this week, Philip Hammond, the foreign secretary, said Russia’s “aggressive behaviour” posed a significant threat to the UK, adding: “It is no coincidence that all the agencies are recruiting Russian speakers again.”
The lulz! They are so strong! So in March we're being told that Brit intelligence is "recruiting Russian speakers again" and in June we're being told that we have a vast network of spies and assets in Russia that are being "moved" instead of being arrested. Again this whole crap beggars belief. How exactly is MI6 supposed to run a spy network inside Russia if it didn't even bother recruiting Russian speakers until two months ago?
No, the more I think about this, the clearer it gets. This story is garbage.