Chess evolved slowly but also mutated quickly at some points in history.
Mutation was due to the influence of another game.
An example of mutation is the Oriental games that were influenced by the oriental game GO
These are Chinese / Korean / Japanese
An example of a branch that did not mutate and is most similar to the origin is South East Asia
These being Thai Makruk / Cambodian and Burmese (I prefer that name)
Burmese did actually mutate and Cambodian (Khmer) is a derivative of Thai Makruk
And now the part that all the "experts" missed
In Europe there was a game slightly different in each country that I call the Scandinavian or Viking game, google Ancient Viking Board Games, or something like that to find out more.
The important point of these games was that some of the boards were large and some of the squares were coloured, in other words the first partly checkered boards.
In the 12th century Persian chess on a 8x8 plain board mutated with the Scandinavian game somewhere in Europe and COURIER chess was born on a 12 x 8 board.
Queen moved like Fers (one step on the diagonal like Thai Makruk)
Ancient bishop moved like Indian, 2 squares on diagonal jumping over the first square
One extra piece that moved like King but was not royal (not checked)
One extra piece that moved like the stones in the Scandinavian game (one step up or down, left or right)
And important, 2 pieces, the COURIER, that moved like modern Bishop.
So you had the old bishops and the new bishops in the same game.
Also important in the starting position a central pawn and the pawn on each file on the edge of the board was already advanced 2 squares, in the starting position, the first occurance of a pawn double step.
And finally the most important change, a checkered board, influenced by the Scandinavian game and needed due to the introduction of the Courier (the modern bishop)
Then later somebody made a 8x8 checkered board, threw away the ancient bishop, threw away the piece that moved like a king, threw away the viking stone and you almost have modern chess.
Double pawn steps then later introduced at first only for the 2 central files (d and e) but later for all files.
Then finally about 500 years ago a powerful Spanish Queen was inaugurated and somebody combined Bishop + Rook = Modern Queen.
Originally called The Enraged Lady of Chess (French), Queens Chess or Mad Queens Chess (English) now simply called Chess
Note that Japanese (Shogi) also has a modern bishop (only one) but you can drop a captured unit back into the game so can have a bishop pair but board is 9x9 and not checkered.
Not sure if the Shogi Bishop influenced Courier Chess, probably not, will never know.
Conclusion
To answer your initial question, all of these changes in Europe then made their way back to the source, India, by way of trade - just like the original source travelled from India to say China along the silk road, etc, etc and boom you have your Indian variants.
Yes I know that all of you are "experts"
Warlord
Final Wars
Chaturanga and shatranj do not have queens nor bishops - they have ferzes and alfils.
When did queens and bishops appear?
By 19th century, they existed in India. For example, the Indian game of Maharaja and Sepoys seems to assume that the Maharaja, an Amazon, faces the army of chessmen including queen and bishops. With optimal play, the sepoys will checkmate the amazon.
If a Maharaja is confronted by Sepoys who did not include Queen nor bishops, just ferz and alfils, does the Maharaja still lose?
Also, into 19th century Indian chess did not have double initial move of pawn. Moheschunder Bannerjee and other Indian players accordingly favoured fianchetto in opening - occupying the centre with pawn moves would be slow.
But this is most favoured if bishops as long range pieces exist to command the centre.
Would fianchetto work with alfils? How should alfils be deployed in opening?
When did Indians adopt queens and bishops rather than ferzes and alfils? And where did queens and bishops first appear?