The Linux developers had a head start in the form of the KAISER (later KPTI) patch set, development of which had AFAIK started before Meltdown was discovered and reported in private to Intel.
After a quick web search, I found https://patchwork.kernel.org/patch/9712001/ which records the initial submission of the KAISER patch set at 2017-05-04. The repository at https://github.com/IAIK/KAISER has an older version of the patch set dated 2017-02-24, indicating that work on it had started even earlier.
Finally, the timeline at https://plus.google.com/+jwildeboer/posts/jj6a9JUaovP mentions a presentation from the authors of the patch set at the 33C3 on late 2016. Note that this page puts the submission of the KAISER patch set at 2017-06-24, but I believe that to be wrong; searching the web for "[RFC] x86_64: KAISER - do not map kernel in user mode" finds several mail archives with that message, and they all agree that the date was on May, not June.
That is, even if Microsoft had been immediately warned by Intel (or by Google), the Linux kernel developers would still have had a few extra months of head start, by basing their work on the KAISER patch set. Was it luck, or a side effect of the Linux kernel being used for academic research?
They said discovered AND reported, not just discovered. It is entirely possible someone discovered it much earlier and didn't report, but we won't ever know if evidence is never found.
KAISER is developed to mitigate another less severe vulnerability.
From the meltdown paper:
> We show that the KAISER defense mechanism for KASLR [8] has the important (but inadvertent) side effect of impeding Meltdown. We stress that KAISER must be deployed immediately to prevent large-scale exploitation of this severe information leakage.
