Update: The Euro-Atlantic Disaster Response Coordination Center which is an instrument that NATO has together with some former Eastern Block states is working on corona virus aid now.
According to German newspaper FAZ the mechanism was triggered by Ukraine on March 23rd (followed by Spain, Italy, ...).
Again, I'm not sure how the aid labeled in particular cases, e.g. whether the German air force who now fly some ICU patients from Italy to German hospitals are counted towards this NATO program or whether this is direct assistance.
Side note: That news post also says that needs Italy posted cannot possibly be fulfilled, citing e.g. that 58000 respiratory machines are needed. To put this number in relation: that's about 3 x the "normal" number of ICU beds with respiratory machines of the whole of Germany (1.4x as many inhabitants as Italy), and twice as many as the current "ramped up" number (source, also explaining that the increase includes borrowing such machines from anaesthetic recovery rooms and re-activating old machines).
Short answer: There are AFAIK several relevant processes within the EU, but they are not labeled as specifically for Italy. Also, some of them provide channels between member states without being very centralized at the EU. Others are clearly EU centered.
One is a concerted buying of supplies, where some member states pool ressources they each buy and a committe in Brussels decides who can when get how much. (Not sure how much they've managed to buy so far since the Covid-"branch" of that program just started recently, and I also don't know which of the EU member states participate),
The second is a civil protection mechansim which was triggered by Italy and Spain. With this, member states build reserves of supplies and machinery, and countries can ask to get some via the EU mechanism. However, as I understand this, supplying a member state is subject to availability and according to this newspaper post so far nothing has been supplied to Italy or Spain via this mechanism.
The EC is buying supplies, trying to leverage their greater negortiation power.
25 member states and the UK participate in this program.
These processes have been in the German and European media as having a very difficult start. Part of this struggle is that Covid-19 relevant supplies have been scarce since before the situation escalated in Italy. Not only in Italy, but throughout Europe if not world-wide.
Germany has delivered supplies (1 mio masks also the ICU ventilators in o.m.'s answer - but it is totally unclear to me whether this was a direct thing between Germany and Italy, or via one of the EU mechanisms (apparently not, since the newspaper says nothing has been delivered via the EU mechanism that could work already) or whether this is a direct thing between Italy and Germany that came off the EU video conferences of
the health ministers or the heads of state.
The FAZ from 23rd (link below) says that PR/media-wise, the German help to Italy happens in a very silent fashion, compared to, say, Chinese help.
I can add that I found it comparatively hard to google newspaper posts for the aid to France/Italy which I remembered to have met during the last days. Sometimes only one bigger and maybe a local newspaper came up, whereas there isn't any difficulty in finding big critique about not helping Itlay in the national press.
I guess, EU not working and China/Russia/Cuba helping is better click-bait than piecemeal inner-EU help for Italy.
Situation throughout Europe
Even though the situation is worst in Italy (at least within Europe - I'm not sure whether Iran is doing better, even though they have substantially lower cases per million inhabitants and also a somewhat unusual linear growth [suspicion: possibly limited by constant daily testing capacity?]), many countries throughout Europe are at best only around one week behind Italy in terms of confirmed cases per million inhabitants (France, Germany, Belgium, the Netherlands), Austria and Norway maybe around 5 days, Spain only 2 - 3 days, Switzerland maybe 1 day, some small countries such as Luxembourg and Liechtenstein already have more confirmed cases/mio inhabitants.
All these by now have more confirmed cases per mio inhabitants than Iran.
While not (yet?) at the limit like Italy, all these countries are in emergency mode as well, and their health systems are very, very busy trying to prepare for the cases that will arrive in the hospitals very soon.
In North Eastern France (Alsace) hospital capacities are also exceeded by now.
Some patients are transported to ICUs in neighboring German Länder for treatment according to ZDF (German Public TV). One doctor explains that they predict they won't need these ICU beds for during the next 7 days, and by then they expect the patients to not need ventilation any more: either they are dead or they are well enough to not need intensive care any more.
A few Italian ICU patients will be flown to Saxony (one of the so far less affected regions in Germany) (Focus).
Side note: these beds being available now does not necessarily mean they have been available a week ago: German health system is currently ramping up capacity for Covid-19 patient needs, elective surgeries have been canceled to free more beds, and the influenza season seems to be about over. All 3 factors contribute to increased hospital/ICU capacity in view of what is expected to happen during the next days/weeks.
Both news posts indicate that they react to official French/Italian help requests, but I have no idea whether these official channels are on EU level, side effects of EU discussions of e.g. the national health ministers, directly between countries or even directly between the involved regions.
(Ir)Relevance of NATO
I don't think NATO is relevant here. While Corona has been called a "slow-motion natural disaster" the usual preparations for natural disasters (beside the recommendation that everyone should have the necessary supplies for at least 10 days at home) are of no particular help for the epidemic/pandemic. They do have medical supplies, but they are mostly oriented towards accidents/injuries rather than a pandemic.
I was wrong, see update above.
China, Russia and Cuba
IMHO China is the "natural" supplier for face masks etc. at the moment: they are usually the world-largest producer, and while their production was severely down and they even became net importers of masks during the epidemic in Hubei they are now again net exporters AFAIK.
As the [first wave of the] epidemic seems to be pretty much over in China, they are pretty much the only country that has doctors right now that have experience with Covid-19 and are not urgently needed at home.
Cuba right now has 40 cases in total, 4 per mio inhabitants. That's like Italy ≈5 weeks ago. Cuba has the world highest density of medical doctors: 82 per 10000 inhabitants (Italy, Germany, Spain, Switzerland, Russia have all ≈40). Cuba has a history of sending medical doctors to help internationally. These services earn cuba approximately 4 times as much money as tourism FAZ Mar 17
Russia currently has 3 cases of Covid-19 per mio inhabitants (438 in total).
This means, compared to EU countries (and North America), these 3 countries are just now in an objectively better position to send help, and of course politically that provides also a possiblity to rub in that the Western World is not invulnerable/pay back Western arrogance.
A rather acidic comment (in Italian) (longer summary in the answer on skeptics.sx) points out that the (first) Chinese aid deliveries were accompanied by extremely clever PR. Including creating the impression that supplies are donated that are actually bought. (Meanwhile, there seem to have been Chinese donations as well)
Not sure whether this plays a role here: after 2nd world war, the Italian Communist Party was the largest communist party in any western country.
(Maybe some Italians can comment, in any case, I've seen 1st of May demonstrations in Italy of a size and frequency I've never seen, say, in the former GDR after German reunification [and for sure not in Western Germany])