I'm not exactly sure how this would help in post-conviction proceedings in the UK (even in the US I'm only moderately knowledeable on post-conviction proceedings), but from a US perspective this could raise significant constitutional concerns.

It's not actually the fact that the drafts were deleted or not deleted that is significant -- it's where does the information truly come from. If Yahoo is right and it was able to recover snapshots because the defendant didn't delete them fully, then it doesn't help at all. If Yahoo is actually lying about it's program and it was able to collect the snapshots because it has been systematically saving all of the mailbox info to its servers in real-time, that would be interesting but it still wouldn't help the defendant.

But, if Yahoo is not being forthright and the drafts actually were deleted from the mail server, it suggests that the evidence must have come from somewhere else. In that case, the most likely candidate is a government entity, such as the NSA (or GCHQ in the UK). If the snapshots are coming from a government entity, which is essentially using Yahoo to hide the fact that it is collecting real-time info on the defendant's email, there would be a number of avenues that could impact the verdict. There could be an appeal based on government misconduct by failing to disclose the information during discovery (it's Brady material). If a government witness relied on the documents for information used in the prosecution, it's probably Jencks material. If the government presented any witnesses who testified that the material was gained from Yahoo when it actually wasn't, this could be Giglio material. If the court finds that the government had an obligation to produce that material and did not, the conviction could be overturned (again, if this were being analyzed in a US court -- I'm not sure what impact, if any, it would have on a UK conviction).