This answer is about Yiddish. There have been various comments, which I build on and add to, as there are several complications:

1. Does it result from a "modern migration"?
2. Is Yiddish spoken in Europe?
3. Is it an Indo-European language?

Let us look at each of these.

1. Does it result from a "modern migration"? Now that the question has been clarified with a link to a definition of modern, the answer is clearly no. So it is eligible.

2. Is Yiddish spoken in Europe? Yes, it has been for centuries, although it is rapidly dying out. So it should be counted as a European language

3. Is it an Indo-European language? The quick answer, according to most sources on the internet, is that it a Germanic language (thus IE) with Hebrew (thus Semitic) added.

But I don't think it is fair to classify any language as IE or non-IE based solely on a simple majority. If there are significant elements from both IE and non-IE, then it is linguistically important as both an IE and a non-IE language, in my view. If we were to go with a simple majority view then perhaps the UK should be excluded from Europe based on the fact that 52% of the population does not want it included in Europe.

When we get to the question of how much Hebrew there is in Yiddish, and thus whether Yiddish makes a significant contribution to non-IE European language, I came across a problem. No easy-to-find online source in English told me. This reflects the low status of Yiddish in the English-speaking world in the 21st century.

I turned to French Wikipedia which stated that the vocabulary is 10-15% Semitic. I would say that this figure, by itself, means that Yiddish should be included, as it means there is a Semitic element in European language, even if it is not large.

But there is a much more important consideration: whereas Maltese has had no significant effect on any other European language that I can find (notwithstanding this list of words I have never heard of), Yiddish has been the conduit for a number of Semitic words to enter not only German and Polish, but also English (paying attention to those marked as Hebrew in origin) (many of which I have heard of) and French.

So, given that Yiddish is the European language which has introduced many (and possibly the most) Semitic words into English, German, Polish and French, I think it deserves a place on the list, regardless of its predominantly IE grammar and the particular percentage of words of Semitic origin.

According to Dr. Seth Lerer, of The Great Courses and University of California San Diego, the Georgian language 's parentage is unknown, so it may not be Indo-European.

I think Kalmyk has not been mentioned yet. And depending on what you define as "late modern", Chinese (e.g. in Liverpool) may also count.

This answer is about Maltese. There have been various comments, which I build on and add to, as there are several complications:

1. Does it result from a "modern migration"?
2. Is Malta in Europe?
3. Is it an Indo-European language?

Let us look at each of these. The main reference is Maltese language.

1. Does it result from a "modern migration"? Now that the question has been clarified with a link to a definition of modern, the answer is clearly no. So it is eligible.

2. Is Malta in Europe? It is an island between Europe and Africa, so, logically, it is neither European nor African. You could say it is European, based on proximity, but as T.E.D. quotes. "Islands are generally grouped with the nearest continental landmass...there are some exceptions based on sociopolitical and cultural differences... Malta was considered an island of Northwest Africa for centuries" so there are arguments on both sides. As for the argument about whether it is culturally European, there are two complications here. One is that it is not h*m*geneous. The northern part (nearer Sicily) is more Italianate (even with quite a lot of bilingualism) compared to the south. The other problem is that not all of its European-ness results from its proximity to Italy (which would help to class it as European) but rather from the fact that the UK treated it as a colony for quite a long time. We do not generally count places as part of Europe just because the UK injected culture from afar. As far as religion is concerned, Catholicism does not make a place European. The Maltese do seem to treat themselves as entirely European (in my limited experience) and they are in the EU, so I would put them in Europe on balance.

3. Is it an Indo-European language? The above reference says it is a Semitic language, but the fact is that it is a creole, with Sicilian Italian as the acrolect and Sicilian Arabic as the basolect. Whether you classify a creole according to the acrolect or the basolect is more to do with politics than linguistics, since the linguistic fact is that it is a mixture. As recently as the 1990s Maltese children were being taught in schools, heavily influenced by the Catholic Church, and on the flimsiest of evidence, that Maltese was based on Italian. Every creole is different in terms of how much of the acrolect and basolect there is in it, but there is typically a higher proportion of the basolect in the grammar and basic vocabulary than there is in the advanced vocabulary. In the case of Maltese the above reference says

The original Semitic base, Siculo-Arabic, comprises around one-third of the Maltese vocabulary, especially words that denote basic ideas and the function words, but about half of the vocabulary is derived from standard Italian and Sicilian; and English words make up between 6% and 20% of the vocabulary. A recent study shows that, in terms of basic everyday language, speakers of Maltese are able to understand less than a third of what is said to them in Tunisian Arabic, which is related to Siculo-Arabic, whereas speakers of Tunisian are able to understand about 40% of what is said to them in Maltese.

That looks to me like a substantially Indo-European language, but also like a substantially non-Indo-European language. Since the question is about non-Indo-European influence in Europe, which does occur in Maltese, I think Maltese should be included, for the Semitic half.

So, on balance, I think Maltese should be included, as it contains a significant amount of Semitic in somewhere that is substantially European.