I don't know if I'm the best person to speak of this - I can talk about their role (and/or lack thereof) in the Indo-European tradition, but I'm going to have to be a bit handwavy with later developments.

So, it's important to distinguish between "Hinduism" as we conceive of it today, and the underlying Vedic tradition. To simplify grossly, the Vedic tradition, especially the Rgveda, is immediately recognizably Indo-Iranian and by extension, Indo-European, whereas the Hindu tradition underwent both internal reform and substantial influence from Dravidic (Telugu, Tamil, etc) traditions.

First, Shiva. Shiva is actually a significant deity in the Rgveda, but he is better known as Rudra there. Rudra is a very recognizably Rgvedic deity - a highly martial, elemental being, associated with storms and the wind. Rudra shares some distinct features recognizable as precursors to characteristics of Shiva, namely an ambiguity or duality between being a fierce and martial being, and a kind and tranquil one (Rv 10.92 - I believe the relevant word is "auspicious" in line 9.2) - the latter is in fact where the epithet/name "Shiva", meaning roughly "kind", comes from. As you can see, that verse actually abounds with Indo-Europeanness - it mentions the thunderbolt-mace (vajra) of Indra, sometimes (in a bit of a reductionist manner) called the "vedic Zeus", and the trifecta is completed with mentions of Mitra and Varuna, gods of the Covenant and Oath, significant to understanding Zoroastrianism. But even more we have mention of Dyaus, the Sanskrit linguistic parallel of Zeus. We also have a mention of "Bhaga", "prosperity", which is recognizable if you have studied Old Persian texts, where "baga" is used idiomatically to mean "god", probably in the sense of "benefactor" (for example, the principal epithet of Ahuramazda in Achaemenid inscriptions is "baga vazraka", where "vazraka" means "great" or "mighty", a descriptor also applied to the monarch - Kshayathiya Vazraka or Great King).

So while I don't know of a specific equivalent of Rudra or Shiva in any other tradition, it is clear that he fits into the context of martial-elemental beings that is so characteristic of the earliest Vedic tradition.

Vishnu is decidedly trickier, as a relatively minor Vedic deity that would rise in profile. Rv 1.154 gives Vishnu the epithets "Bull" (also applied to Agni), "prowling", and "mountain-dwelling", and bring up his "three steps" in which he covers all creation. He's also quite clearly identified as a maintainer or upholder of the world, one of his most significant characteristics in later Hinduism. (It's hard not to get associations with Atlas, with upholding the world, having an immense reach, and being associated with mountains - but this may be spurious and is my speculation alone). Interestingly, Rv 1.56 describes similar characteristics and appears to be attributed as a hymn to Vishnu by some - but on a careful reading, the text seems to be about Indra (though the hymns can be confusing, this seems clear when you consider "soma-drinking" in the first line), and the word "vishnu" is never mentioned. But it's clear that there has been some conflation when you compare to 10.56, where Vishnu is associated with Soma-drinking:

MAY they ascend, the lowest, highest, midmost, the Fathers who deserve a share of Soma-

May they who have attained the life of spirits, gentle and righteous, aid us when we call them.

Now let us pay this homage to the Fathers, to those who passed of old and those who followed,

Those who have rested in the earthly region, and those who dwell among the Mighty Races.

have attained the gracious-minded Fathers, I have gained son and progeny from Visnu.

They who enjoy pressed juices with oblation seated on sacred grass, come oftenest hither.

(On Soma, see /u/artfulorpheus answer here. It's a substance of ritual importance, apparently psychoactive.) Vishnu seems to be given a creative or progenital role here. However, Vishnu's name is mentioned that once, and the later half of the hymn goes on at length about... Agni.

This starts to get almost odd when you look at Mandala 7.99, where Vishnu is described in incredibly exalted, even superlative, terms, and appears to be conflated with Indra. He's also given credit for spawning the trinity of Surya, Ushas and Agni (lit: Sun, Dawn and Fire) - Surya occurs in various couplings with other significant deities as well. I am absolutely not a Vedic scholar, but this verse looks a lot like a later insertion to me. Verse 100 goes on along these lines:

Foremost be Visnu, stronger than the strongest: for glorious is his name who lives for ever

and so forth. These occur right at the tail end of what is supposed to be the "core" of the Veda (Mandalas 2-7), centuries older than Mandala 1 and 10 (Mandala 8 and 9 are too inconsistent to be given a general date). I'd love to hear from someone who is a Vedic expert to opine on this, but it looks like these verses, being so superlative, might have been inserted to give support to later verses like 1.56 being understood as being about Vishnu.

So, what we can see with Vishnu is a pattern of ambiguity and occurring alongside other more significant Rigvedic deities, which sometimes seem to evolve into Vishnu outright eclipsing them or being considered their creator. The name Vishnu is usually translated as "the all-pervading one", i.e. the one who is present in everything, and this seems to fit into this pattern well (the literal meaning should simply be "pervasiveness", "vish-" "to pervade" + "-nu", which forms nouns in this manner). Vishnu, thus, is an ambiguous entity pervading all sorts of aspects of the divine - and this characteristic in turn makes it easy to understand how such a being could be exalted into a supreme or near-supreme being by Brahmin looking to form a cohesive interpretation of the text.

Now, the Rig Veda dates to, broadly, the middle of the second milennium BC, plus/minus a few hundred years depending on the verse and mandala. I am heistant to opine on the later Vedas and much of the later scripture (but broadly, Vishnu is described in exalted terms in e.g. the Narayana Sukta), but by the time of the Bhagavad Gita (dated to around 500-200 BC), we find the familiar Krishna-as-Vishnu's avatar story (the notion of appearing as an avatar seems to follow from Vishnu's "pervasiveness" and general manifoldness), as well as arguably the beginnings of the Bhakti movement, and over the next milennium the gradual development of Vishnuism (or Vaishnavism) - and ultimately Hinduism as we think of it.

To summarize, then, Vishnu seems to have started out as an obscure and ambiguous Indo-Iranian deity that gradually grew in importance by conflation with and exaltation over much more significant Rigvedic deities like Indra and Agni. I am not terribly surprised that we don't find any analogues in other Indo-European religions - Vishnu is a typical example of a highly abstract concept (pervasiveness) being personified as a deity, and this seems to have occurred more in Indo-Iranian religion than in the west-lying mythologies (though we also have, comparatively, a vanishingly tiny amount of early evidence on them, so it is hard to say for sure). There are tons of such hypostatic deities, not all of which are always translated as proper names (I'm sure there might be instances where Rgvedic texts would actually read better if "vishnu" is read literally as "pervasiveness", but who knows - I'm not nearly fluent enough in Sanskrit for that!). While "pervasiveness" might not be a familiar deity from other mythologies, it's possible that it was understood as such at some point and simply grew obscure, rather than, as in Hinduism, being exalted.

I hope that answered your most pressing questions - I have to run now!

I have relied here mainly on the Rg Veda as a primary source; my main basis for the context are my main references on Zoroastrianism:

• Wiley-Blackwell Companion to Zoroastrianism (especially the chapters about its Indo-Iranian roots).
• Zoroastrians: Their Religious Beliefs and Practices by Boyce (first three chapters include significant discussions of the Vedas).

I've also read bits of a book called Aspects of Early Vishnuism once upon a time, which provided some useful philological context.