Oh dear. The translation of the title of Plato's Πολιτεία as "Republic" is a misleading one in English. The state (or rather, states, as people tend to forget that Plato goes through a number of possible cities) described in the Republic is, of course, not a republic at all by any reasonable definition. Instead, medieval scholars translating Plato into Latin translated πολιτεία, meaning roughly "city matters," as Latin res publica. But res publica doesn't mean "republic" either, in that it does not describe a particular form of representative government the way our modern English word does. Res publica simply means "public matters," roughly analogous to πολιτεία. The Romans used the term of all civic institutions following the expulsion of the Tarquins, and continued to describe the state as res publica well after even they had acknowledged the last republican institutions to have died out.

We have Plato pretty much in his entirety, by fortunate circumstance, and if Plato knew about the city of Rome he never mentions it. Plutarch says that Aristotle a generation briefly mentioned the sack of the city by the Gauls, but there's no reason to suppose that Plato knew anything about Rome. Moreover, it's unclear precisely when republican institutions replaced the kingship at Rome. The 510/509 date is a later interpolation, invented to connect Rome to the expulsion of the Pisistratids at Athens in the same year--in reality the kingship probably survived at least another generation if not another century. Dating is hard. And none of the states in the Republic bear much resemblance to the Roman Republic at all, as a quick look at Plato shows instantly. The Romans certainly did not know any Plato, besides possible rumor of his existence. The first major philosophical school to enter Roman society was Epicureanism, which made quite a splash when it was first introduced--from the Epicureans the Romans learned of the Stoics, Academics, Peripatetics, Cynics, and all the other major Hellenistic schools.

The two have nothing to do with each other

Or did Socrates take his ideas from practices in place in the Roman republic. Were Socrates and Plato essentially promoting Rome over the Athens democracy? Would Socrates even consider Rome a 'true' republic?

This isn't related to anything I said above, but let me make it absolutely clear: Plato's character named Socrates and the historical Socrates are not the same person. Socrates wrote nothing down, and we know from the tradition that the various Socratics wrote Socratic dialogues like Plato's, with the character of Socrates differing in each of their treatments. The Socratic Question, the determination of just how much of the historical Socrates is actually in Plato's character, is among the most fraught topics in philology, and the balance of opinion is generally that it's all but impossible to muddle out the real Socrates from Plato. In some cases it's quite obvious that what Plato puts in his character's mouth is not the opinion of the historical person--the majority of the Republic probably falls into this category. The Republic is not really a dialog, it's really a treatise in all but name, and Socrates' behavior throughout it is decidedly unlike his behavior in the so-called Socratic Dialogues, which are apparently closer to Socrates' own behavior. There is, generally speaking, little hint of the character's views from earlier dialogues except where we can conjecture that Plato expanded on some influence from the historical Socrates. Socrates' opinion on government is lost to us, and Plato's is very hard to figure out--the Republic confuses itself partway through, and at times Plato seems to be talking about psychology rather than government

EDIT: On rereading the question, looking over your reply prior to its removal, and looking at /u/Polybius' comment I want to make sure that I'm not misunderstanding something. You mentioned Cicero in your reply, which seems to me to have no relevance given that I've understood you as asking whether Platonic thought had any influence on the creation of the Republican state, which I think I've established cannot reasonably be shown. But are you in fact asking about Platonic influence over the course of the entirety of the Republic?