How did Brahma, Vishnu and Shiva come to replace Indra as the Chief of the Hindu pantheon? Was there a war (e.g. Titans vs Olympians)? Or were they consumed (e.g. Amun + Ra = Amun-Ra)? What changed and was the cause of the change in the Hindu mindset?
In Hindu mythology, Trinity was always at a higher level than the lower deities like Indra etc. Vedas recognize the trinity as being superior and there are numerous examples throughout mythology where Indra and others used to reach out to the trinity to help with their problems (usually with some demons etc that got too strong for Indra and others).
There never was a scenario where the trinity and other deities were at the same level either in mythology or in contemporary culture/traditions and then something came to pass that increased the status of trinity. The reason for this is that, in Hindu creationism trinity come into being first. Then they, using their powers, create rest of the universe including the lower deities. The exact process by which the lower deities come into being varies but the important thing being that they are created afterwards and therefore are lower in status. We humans, Indra, rest of the pantheon, demons, what have you can be considered children of trinity (for this discussion).
Regarding the mindset of people, in the vedic times it was a common practice for people to worship both trinity and the lower deities. Trinity were still considered to be at a much higher level but the lower deities were also given their dues. Some of it happened because people feared the lower deities (like losing their crops if Indra gets upset and brings a flood etc, which he could and did) and some of it was just tradition. In summary, the trinity were always considered separate and a much higher entity than the lower deities both in mythology and contemporary culture, but the lower deities were still worshiped as well.
Now, to answer your second question, what caused the people to eventually stop worshiping the lower deities (specifically Indra) - If we go by the mythology, this began changing in Treta Yuga when Vishnu incarnated as Krishna. Krishna actively dissuaded people from worshiping lower deities (and any other local gods, forest gods, tree gods, what have you) and exerted strong influence to move people towards monotheism. Gradually people stopped worshiping the lower gods/deities and when Indra got upset and tried to punish people, Krishna protected them from his wrath very publicly to prove his superiority.
Then in Geeta, Krishna effectively sealed the deal by explicitly stating that no matter who you worship, you're effectively worshiping the one God, even if by proxy. He also stated that He himself likes the worshipers who understand His true nature (that is, one God) and the worshipers who worshiped him directly got better results than those who did it by proxy. This coupled with Krishna publicly showing people that He can easily protect them from Indra (the king of gods) if they don't worship Indra left no reason for the people to worship any of the lower deities
There were no "fights" between Trinity and lower deities simply because trinity would simply destroy Indra and others and replace them with their newer versions in a blink of an eye (Shiva literally did that with Kama, no pun intended, when Kama tried to go against him).
Finally, the final nail in the coffin was during the last 2000-3000 years or so when Puranas were finalized and depicted Indra involved in various "ungodly" scandals and then more so in Bhakti kaal when Tulsidas and others further popularized incarnations of Vishnu and Shiva (Ram, Krishna, Hanuman etc). Before this people used to worship Shiva, Vishnu and different variations of adi shakti mata but Bhakti kaal resulted in the further popularization of many of the reincarnations as well.
Note: The lower deities aren't completely dead, so to say. They're still worshiped in Hawans and rituals and what not as part of tradition.