I hate to be the bearer of bad news, but the path of becoming a consultant or an adviser in a video game studios, a film company, or a similar position is complicated. Fundamentally, however, they are looking for individuals with a proven expertise in a topic. That means having published original research, authored books, or earned a research degree. In a post from 4 years ago, /u/Iphikrates talks about his own experiences in being part of a documentary (which he has likely much more to add now, judging by a recent popular video on YouTube he's featured in), yet since your post talks specifically about video games, I will focus on that.

In the case of First World War video games, for example, the producers turned to individuals with established credentials. For the game 11–11: Memories Retold used historians Peter Doyle and Rob Schaefer as consultants to add not only their expertise but also to legitimize the content of the game as being historically accurate. The producers learned about them through their work and expertise, in similarity with Iphikrates experience above, and contacted them as a consequence.

However, this is not always the case. Many studios, like Ubisoft or DICE, have their own employees make the research. In Chris Kempshall's research into the historiographical basis for Battlefield 1, he found through interviews with the game developers at DICE that they had already read many books and interviews with historians on the First World War as the game began to be developed. Indeed, Kempshall finds, they were even somewhat determined to move away from older interpretations of the war in line with contemporary research! While their intentions were good, they ultimately partially succeeded in reaching this goal. Battlefield 1 does make use of historical consultants to a certain degree. The developers took advice from Indy Neidell, the host of the YouTube channel The Great War, for the game and also hired him and his research assistant to write the in-game codex. The developers undoubtedly found out about Neidell due to the success of the YouTube channel, which shows another avenue into becoming a historical consultant, but this of course has its own limitations for the gaming company; the end result is that the historical codex of the game was somewhat lacking in certain parts, something that I've worked with in my own research.

To summarize: Game studios are looking for people with a proven or known expertise in relation to a specific historical subject. These consultancies will always be temporary, as game studios do not have in-house historical experts working for them as a full-time job. For the most part, the developers and writers of the game will do their own research -- which as we all know does not always have successful results.

For more on First World War gaming and historical sources, see Chris Kempshall, "War collaborators: documentary and historical sources in First World War computer games," First World War Studies, 10:2-3, pp. 225-244.