Two Saturniidae species have been introduced to Europe. Samia cynthia (mentioned here) and Antheraea yamamai.
Interesting is that both species have been introduced here deliberately, by people who were looking to exploit their populations for silk (sericulture). I think generally, the introduction of such a species requires a constant flow of introductions - the chance it happens from only a few escaped individuals is extremely low, as Rudi pointed out. The biggest threat would be people deliberately releasing animals. Even the famous Lymantria dispar in America was initially introduced by an entomologist who was free-breeding them in his garden - with no net to contain them - as an alternative for silk, until they began to spread.
The good news is that, while a lot of harmful and invasive Lepidoptera exist, generally, Saturniidae are almost never considered 'harmful'; because of their large size and biomass they live in lower densities and have slower generations compared to some other harmful invasive species. This is however an overgeneralisation. Some types of Saturniidae (like Hylesia) can be an agricultural pest; but generally, being able to call a Saturniidae a 'pest' is rare.
Though these moths don't seem to be very agressively spreading ones, so they shouldn't cause significant change to native ecosystem
The problem is; how do you measure this?
Every species you release into a new environment, alters it. Some alter it a lot, but damaging native trees and defoliating them. But on an immeasurable scale, species interactions also change. Maybe it's a new host for a parasitic fly that is otherwise rare, but boosts their numbers? Maybe they become the favorite prey of an insectivorous bird or bat which boosts their populations? They may even bring new micro organisms (diseases) with them and more.
All of this is very hard to measure.
Maybe, some of these changes are insignificant, but they are still there. And they can quickly add up.
What if people introduce an invasive species together with an invasive host plant, and then an invasive parasite is introduced? This is what happened when Compsilura concinnata (Tachinidae) was introduced to destroy invasive Lymantria dispar; they also begun to attach many native populations of endangered American butterflies and moths. Oops.
In that regard, I do consider myself to be a 'purist', and releasing exotic animals should strongly be discouraged. At worst, it will harm the environment, and at best, even species seems harmless - they still change it in some way, since every species has interactions with other species.
In my breeding career I sell a lot of cocoons, but only three times (3) in 10 years did I have to decline an order to a customer, because they revealed their intentions of releasing the moths outdoors. In that case I refuse to sell to that customer in the future. The species were Danaus plexippus, Samia cynthia, and once Hyalophora cecropia.
Note that it can also damage the reputation of our hobby, and get species outlawed.
Much of the outlawed snakes, reptiles and fish in Europe, have been banned because they were proven to breed in the wild once escaped, or are considered to be too dangerous after an escaped animal injures a human. So I also consider other people who release exotic moths outdoors to be a threat to my hobby - and work.
That being said, are there Saturniidae that would survive in Europe? Difficult question. I think the temperature is fine for most species, some come from much colder climates. But more complicated factors are precipation (humidity), light intensity, availability of host plant, et cetera.
I think Antheraea polyphemus would definitely survive in the warmer parts of Europe. Maybe some of the northern Anisota species too. I'd also potentially nominate Caligula japonica in places with a mild winter (Balkans?) and maybe Opodipthera eucalypti in hot, dry places with Eucalyptus (Spain?) since they can also overwinter in New Zealand as introduced species. But who knows, I hope my speculation remains pointless here