The answer to this is: there is no definite answer as of yet, and it's a tough topic I could never address properly here. There is a great deal of debate about the continuity of things (I'm intentionally vague) from the end of the Bronze Age down into the beginnings of what we have traditionally called the Iron Age. The crucial period is from about 1125 to about 1035 BCE, from the very end of the Mycenaean period to the earliest hints of the Protogeometric period. This intermediate gap of time, about 100 years, is often called "Submycenaean" due to the appearance of a type of Mycenaean-style pottery which was still obviously Mycenaean, but in "debased" form (i.e., uglier, more poorly made, more poorly painted/glazed). I won't go in to the topic of the "Dorian invasion" as an explanation for the end of Mycenaean Bronze Age, and much has been said about it on this sub already. For a thorough but compact rundown of the arguments for and against the invasion hypothesis, see Bernard Dietrich, The Origins of Greek Religion (Berlin: De Gruyter, 1974/2016), Chapter IV "The Problem of Continuity in the Dark Age".

At the heart of the debate are two sides: some argue that there was a very obvious and, often, violent event or series of events which seem to have grossly interrupted the Mycenaean way of doing things, and when the dust settled about 100 years later, the people inhabiting the regions roughly later termed "Greece" were doing something different, in many aspects of life, than what Mycenaean people had been doing. There was, therefore, a break in the continuity of "civilization" in all its aspects, and then a resumption thereafter. The other side acknowledges that yes, there were violent and poorly understood events happening at the end of what we traditionally term the Mycenaean period, but they argue that there was not necessarily a break in continuity of practices of all kinds, just natural internal adoption of new ideas/methods/techniques/whatever. Each side has done a lot of cherry-picking of evidence, and the body of evidence we have, especially the archaeology, is very ripe for cherry-picking because it is confusing, the nature of the evidence changes from place to place, and is difficult to find consistency in that evidence which reaches every nook and cranny at every time.

I'm getting to the religion, but first I want to talk about pots. The bulk of my experience in this era is archaeological, and particularly ceramic. Pottery provides a great case study of the divide in opinion on Bronze Age to Iron Age continuity. In my humble but experienced opinion, there is a clear break in Mycenaean pottery technology, style, shape, and decoration across this divide, especially when it comes to figural elements. To others (like Dietrich argues), one can perfectly trace a continuous internal development from Mycenaean -> Submycenaean -> Protogeometric -> Early Geometric. A break does not mean, for me, that mysterious invaders had come in, burned everything, murdered all the hapless Mycenaeans, and set up shop, however. The question might be: who is making the Submycenaean pottery? Clearly not the same people, and I don't think that's up for debate. If it was the same people, we have to answer the question as to why they are suddenly doing it so very differently, in different places, with different materials and different techniques and different decoration. If not them, then who? I don't myself care as much about this question. I don't care if it is scary Dorian barbarians from the wilds, or else regular old "Mycenaeans" who have found themselves in changed circumstances, maybe forced to make pots without the tutelage and advice of old guys and suddenly having to wing it as best as they could. The bottom line for me is that there is, in virtually every observable aspect, a very clear breaking point which produces a progression from late Mycenaean to Protogeometric. Such a breaking point is never so obvious later, in the progression from Protogeometric to Early to Middle to Late Geometric, and then to Early Protocorinthian to LPG to Corinthian and then into the Archaic period. Yes, there is definitely a problem of quantity, geography, and provenance of evidence. We just don't have very many examples of Submycenaean.

Now on to cult practice. There is already a gigantic study on this very topic: Marakas, Ritual Practice between the Late Bronze Age and Protogeometric Periods of Greece (Oxford Archaeopress 2010). She studies ritual sites in (the area later termed as) Greece from the Late Helladic III B period (1340 to about 1185 BCE, firmly Mycenaean) to LH III C (about 1185 to about 1050 BCE, including the long period of "collapse" and the Submycenaean) to the end of the Protogeometric (about 1050BCE to about 900 BCE). She uses a system of categorization to identify and classify cult sites: Principal Criteria (assemblage of votive offerings; present of cult statue or image; "religious iconography"; evidence for ritual dining (pottery and animal bones); and Minor Criteria: built furnishings for ritual action; a hearth area (including ash altars); features for libation practices. She further identifies three types of cult sites: those in or associated with palaces; those in settlements (not in a palace); and isolated shrines which were out in the middle of nowhere and not associated with any sort of population center. The study is fairly meticulous. To grossly and inelegantly summarize: the people in the Protogeometric did a lot of the same things that the Mycenaeans had done: they left votive offerings to the gods, though not nearly as many as before; weirdly, there is not a lot of evidence for libation (ritually pouring out wine) in the Protogeometric, but tons of it for the LHIIIB (but we know the Greeks did libation later); they indulged in ritual meals (both drinking and also eating sacrificed animals), and they increasingly did this in communal settings, perhaps in imitation of old Palatial practices; but animal sacrifice did not seem to be the central activity in the Mycenaean period; related, there is increasing evidence of the encroaching domination of ritual activity by social elites, which was not apparent in Mycenaean times, and the act of sacrifice is very intimately related to Iron Age elite prestige display; and, finally, the ritual act of procession (like a parade to and from a spot) was present in Mycenaean ritual and persisted down to the Greek Iron Age. On this last, De Polignac has very convincingly argued that the ritual procession was very important in defining the physical boundaries of emerging poleis. And finally, this study showed that the "isolated" shrine type of the Mycenaean period had typically been in hard-to-reach uninhabited places, but by Protogeometric they had been relocated down in more accessible areas, almost always on top of the ruins of old Mycenaean settlements. This seems to Marakas to be clear evidence that those coming later wished to reacquaint themselves and their ritual actions with a lost but not forgotten Mycenaean past.

As for the gods themselves: it is difficult to say precisely if and where their specific identities were established. It is beyond question that some familiar names of gods were Mycenaean, and since reddit can display unicode Linear B symbols, why the hell not: Ares (𐀀𐀩 a-re), Athena (𐀀𐀲𐀙 a-ta-na), Zeus (𐀇𐀸 di-we / 𐀇𐀺 di-wo), Artemis (𐀀𐀴𐀖𐀳 a-te-mi-te), Hera (𐀁𐀨 e-ra), Hermes (𐀁𐀔𐁀 e-ma-?), Poseidon (𐀡𐀮𐀅𐀃 po-se-da-o), etc. I am not an expert on Mycenaean religion and I don't know how exactly these gods compare across the periods, however.

For some expansive discussion of this very difficult topic, let me first recommend Robin Osborne, Greece in the Making. It is a fantastic book all around and just about everyone can and will enjoy it in some capacity. Specifically on religion, Alcock and Osborne's edited volume Placing the Gods is great, especially the early entries in there by De Polignac, Peatfield, Wright, Antonaccio, and Morgan, which are all important and frequent names for this topic. I mentioned Dietrich, Origins of Greek Religion up above. I wasn't able to read as much of it as I'd like due to the online access constantly glitching out, and some of it felt a little dated, but it is still obviously a go-to. Those are good places to start, at any rate.

edit: I wrote LH II instead of LH III twice (:facepalm:)