First of all, Saturnalia has no direct relation iirc to solar deities or the dying-rising mytheme. I think you are confusing this festival (which honored Kronos/Saturn) with the fourth century Roman Natalis Invicti feast which certainly was a solstice solar festival (honoring Sol Invictus). Both may have had a cultural influence on the observation of Christmas in the early Church, but that's a separate issue from whether the narrative or theological role of Jesus of the NT and gospels draws on non-Christian solar deities like Sol Invictus, Apollo, Mithra, Horus, etc. The "copycat" claims popular on the internet (e.g. that Horus was born in a manger and had twelve apostles and was crucified, etc.) are largely bogus; they invent a "mythology" for the pagan god that cribs from Christianity and bears little resemblance of actual ancient beliefs. Most narrative parallels to details in the gospels lie instead in midrashic interpretation and reuse of the OT — i.e. drawing on native Jewish storytelling resources than pagan ones (e.g. the story of the temptation of Jesus is based on the exodus-wandering narratives in the Torah). Most of the genuine parallels to non-Jewish mythologies are of a more general kind (literary and folkloric topoi), which do not necessarily involve direct dependence. The relevant example of this here is the dying-rising motif. Although this is utilized in solar cults wrt winter solstice, in the ANE it was instead associated with the heat and dryness of summer and the return of rain in autumn. This is what traditionally was the case with Baal, Tammuz, and Adonis in Canaanite, Israelite, and Phoenician religion (the ritual mourning of Tammuz confirmed as part of pre-exilic Jewish practice in Ezekiel 8). These were fertility and/or rain gods, not solar deities (the solar deity in Canaanite religion, Shapsh, was a rather minor goddess). The parallel with Baal is not simply the dying-rising topos but also Baal's role (the adopted son of El, the creator god, who reigns at the consent of his father). However this is also very general as well. But at least we have a direct link: Yahweh in the OT draws on older Baal motifs (along with El motifs) and Jesus was quasi-identified with Yahweh in the NT. The throne vision in ch. 7 of Daniel in particular was of crucial importance in early Christian christology and it is widely recognized as incorporating the older El-Baal dynamic. Still the Christian-Canaanite parallel involves only the most basic themes (father-and-son, dying-and-rising).
As for the mystery cults in the later Hellenistic and Roman eras, it is difficult to know exactly what sort of myths were associated with them since the evidence is usually very sketchy, late, and a lot that has been reconstructed is known to be unreliable. So for instance, some have reconstructed a "Mithras myth" that has strong similarities to the gospel narratives of Jesus, but often this picks and chooses motifs from the very different Iranian (re Mithra) and Roman (re Mithras) cults and much of it is inferred rather than attested. The Isis-Horus cult has solid attestation by Plutarch, but some arbitrarily add motifs from the very long history of these deities, and it is unclear if some of these were present in the Roman-era religion. The Sol Invictus cult itself is quite late, largely adopted after its promotion by Emperor Elagabalus in the third century AD. I will say there is one facet to the narratives of Jesus that I feel has good evidence of at least being parallel to certain mystery cults and mythologies: the child-in-peril theme of the birth narrative. The Lukan birth narrative has no relation to this, but the messianic birth narrative in ch. 12 of Revelation attests a narrative complex found in the Isis-Horus and Leto-Apollo myths (with parallels to the Rhea-Zeus myth): the mother gives birth to a god or savior figure, the child and mother are threatened by a monster or malevolent figure, the mother and child are taken into hiding (by the wind or a bird) in the wilderness or a remote island. The Matthean birth narrative draws virtually all its narrative features from the story of Moses from the Torah and in midrashic retellings, but it shares the same basic plot — with Herod taking the role of the dragon in Revelation, Seth/Typhon in the Isis-Horus myth, Python in the Leto-Apollo myth. That there is a relationship is also seen in the various Illuminator birth legends retold in the 13Kingdoms section of the Apocalypse of Adam (re how the different nations have different stories about how the Illuminator comes), which has clear parallels to ch. 12 of Revelation, the Isis and Leto birth legends, the Mithras birth legends, etc. It is however unclear which came first — whether the Matthean birth narrative historicized the mythological child-in-peril plot from Revelation by drawing on Moses midrash or whether the Matthean story or a precursor of it came first independent of the Isis/Leto myths and John of Patmos incorporated a version of the myth in Revelation on the strength of the parallels. There is a clear thematic connection between the Revelation and Matthew texts, but all the features that most strongly resemble the Isis/Leto myths and 13Kingdoms are missing in the Matthean story, and the OT itself provides sufficient source material (but not combined in the same way; the massacre story draws on Pharaoh's massacre of Hebrew male children whereas the flight-into-Egypt story draws directly on the flight-from-Egypt story of a more mature Moses into the land of Midian). So there is a likely relationship even though its nature is somewhat unclear. But this is the kind of evidence (with a more detailed narrative plot) based on genuine ancient sources that is more convincing than reliance on very general topoi and on very late and questionable evidence (such as a 19th-century self-styled Egyptologist's rather speculative synthesis of the Isis-Horus mystery cult).
