I saw the there are services that allows you to have the possibility to send/receive your emails (we have Office365), even if the mail provider has downtimes (for example one of these services is named "MIMECAST email management).
We have two offices, one in Italy and the other in UK, connected in VPN, and I was wondering: if my O365 mailbox is unavailable in Italy (due to some fails on MS servers in italy), could I reach it anyway going via UK (then without using that mimecast service)?
It really depends on the type of outage. But this is all taken care of Microsoft's engineers as they are the ones running the service on your behalf. And as part of this service you are getting financially-backed SLAs, multiple level of redundancy, disaster recovery and so on. There's tons of documentation available on this online, go and review it, it might help you sleep better :)
Putting a service such as Mimecast in front of O365 mail flow just for the reasons of service continuity is a waste of money IMO, but I'm sure they will be more than willing to take your money :)