The Huns have nothing to do with the Mongols. The Timurids are an off shoots of the larger mongol peoples (with Turkic influence).

The only link between them is that they are steppe people, which is a category as vast as "native Americans" or "Europeans". It's geographic and includes a lot of very different people that have nothing to do with each others.

Timurids are Mongol-Turkic people, so similar origins to Mongols of course. Huns origin is more mysterious however there are also possible links between them, Mongols, Turks, Ugrofinic tribes (Hungarians, Fins).

The Timurid Dynasty and their subjects were ethnically diverse but they were majority Turkic and Persian, basically the same as people in places like present-day Kazakhstan and Tajikstan. Timur claimed to be descended from a cousin of Genghis, while his mother was probably a Turk who spoke Persian or a Persian. One of Timur's ancestors married a granddaughter of Genghis, so his dynasty called themselves "Gurkani" or "Son-In-Law."

The Mongols come from, well, Mongolia. Their language isn't related to Turkic or Persian languages. You'll notice I'm separating people pretty broadly by language groups but back then they probably didn't see themselves that different ethnically in a broad sense from Turkic nomads. There was plenty of intermarriage between Turkic and Mongolic speakers.

Nobody knows 100% sure who the Huns were. They predate the Timurids and Mongols by roughly 700 years, and there isn't as clear written records linking them the way the Timurids and Mongols are. We have hardly anything of their language documented, but based on the names of their rulers they were probably Turkic speakers. There appears to be continuity between them and the Bulgars, the Turkic overlords of the people who would become the Bulgarians, so that's a little bit more evidence towards them being Turkic speakers.

The word "Hun" is etymologically related to Xiongnu (if you ever watched Mulan, that's who the "Huns" she fought are based on), the major confederation of Steppe people in the late BCs and early ADs. That confederation probably included the ancestors of Mongols and Turks, but their language the language of the ruling dynasty doesn't seem to be either. Dynasty names over time become ethnic names pretty often (like how Han is used for ethnic Chinese people but originated from the Han Dynasty), so it's probable that some of the Huns' ancestors took on the Dynasty name of their rulers and eventually considered themselves Xiongnu ethnically, then rode West and South. That's the earliest possible link between them and the Mongols.

The Huns' origins are mysterious. The most commonly purposed theory is that they were the same people as the Xiongnu mentioned in Chinese sources, but even then, it's not known whether they were Mongolic, Turkic, Iranian, Uralic, or something else.

The problem is that the only written sources on the Huns and Xiongnu were outsiders and little of their language has been preserved.