In the case of an EMP it depends on how close you are and how much damage it does. If it is close enough to fry the electronic circuits of the disaster radio, then it's toast. Then again if you're that close everything that could connect to the disaster radio would also be toast. You'd be OK if you kept the setup in a highly shielded bag and only deployed it after the disaster (EMP) struck... however you'd still have the problem that your only device to connect would also have to be stored in the shielded bag.... and it's not much use of a mesh network if everyone else got fried too (because they didn't shield bag storage).
As for Solar storm. I presume you're talking about something like the Carrington Event ( https://en.wikipedia.org/wiki/Solar_storm_of_1859 ). So again, it depends on the scale. Based on the research I've seen, the equipment may keep working - but the radio propagation is a different story. GNSS systems will likely be toast, so that's not going to work at all. Radio propagation, depending on a lot of factors, will likely be affected. If you're in a heavily populated area with lots of notes - it may not make a different at all. If you're out in the fringe areas and only have a few nodes barely within reach - then your mesh might get cut off. However in theory with varying "jamming" by the storm, and retries built into the system, your messages could eventually get through.
If you want disrupt an electrical grid, there's a lot of easier -- and less likely to have an obvious calling card -- than setting off a nuke.
And if you're thinking about some other type of EMP weapon, forget it. They're not practical. While a theoretical weapon, they're not a realistic one. It's fantasy.
https://www.vice.com/en_us/article/kwxq4v/we-asked-a-militar...
https://nationalinterest.org/blog/buzz/world-wont-end-danger...