As a British Indian I have a interest how the ITTL relationship between the British Empire and India would turn out.
Firstly my view is that had India been more developed than it is now then the gap between India and the rest of the Empire in terms of population would be far lower than it currently is.
Secondly retaining the 13 colonies and expanding into the rest of the OTL USA and Northern Mexico as well as maybe annexing Indonesia and Argentina would mean that with a much bigger North America combined with Greater South Africa (i.e if Southern Rhodeshia joined South Africa it would mean that Southern Africa would have a bigger white population overall due to butterflies, as well as a Colored and Indian population which might accept British rule) Australia and New Zealand* as well as the UK and Ireland, it would provide a counter balance to India and thus makes it less likely that a Imperial Federation will become completely dominated by India.
Lastly when you deal with Indian representation you should start with Property and Education Qualification system used in the Cape, thus the franchise will only expands when more Indians get educated or serve in the armed forces. Thus by the time there is a universal franchise (or close to it) there would be a large number of Indians educated under the British which will be English speaking, Anglophone and keen on strong ties with the UK.
So under these three suggestions, India would play a big part in a Federalised British Empire, but they will not be the dominant party.
*All of whom would likely have the "More British than British" mentality.