We should not misrepresent what Laurence Fox actually said. He wasn't criticized for claiming that Sikhs fought exclusively in British Indian regiments, something which he never claimed in the first place. What he did say was that the inclusion of a Sikh soldier in the movie 1917 broke his immersion and that, "There is something institutionally racist about forcing diversity on people in that way." Fox's statements were ridiculous, ignorant, and racist, reflecting what I have termed as the 'White Mythic Space of the First World War'. There is an understanding of the First World War amongst some that the First World War was a 'white man's war' in which people of color either wasn't present, or if they were, they didn't matter. This is something that has been reinforced by popular culture, which often has not bothered to show a representative depiction of the First World War in terms of diversity. Therefore, when encountering attempts at a historically accurate depiction of the First World War in terms of ethnic diversity of soldiers, these people try to actively defend the white mythic space through various means. One of those is by outright denying or belittling the inclusion of soldiers of color. For more on this, I refer to my article "Race, Battlefield 1 and the White Mythic Space of the First World War", Alicante Journal of English Studies 2018, 31: 187-193.

To answer the historical part of your question, the answer is: It depends on the person. Sikhs who were enrolled in the British Indian Army were naturally in their own colonial regiments. However, domiciled British Sikhs would have been in regular British regiments since the 1907 Manual of Military Law allowed men of non-European ancestry to enlist in the British Army. Therefore, British men of Asian and African ancestry were permitted to join regular British regiments. Therefore, the location in which you enrolled played a large part in where you would eventually end up. A man from the British West Indies who enrolled in Jamaica would be included as part of the British West Indies Regiment while his friend, who enrolled in Great Britain, would have been part of a regular British regiment. The regiment you were part of also had a great impact on whether or not you were even allowed to fight. Black British soldiers in regular British regiments on the Western Front would have seen active combat while this would have been denied to the British West Indies regiment.

For the British Empire, there were two main arguments against the inclusion of ethnic minorities in active combat roles. First, and perhaps most importantly, was the protection of the racial hierarchy which was at the very core of British imperialism. By allowing men of color from the colonies, in particularly of African descent, to fight against and even kill white men would upset the status quo which in turn might have larger consequences. The second argument revolves around racial prejudice and the idea that men of color simply did not have the intellectual or emotional capability to be good soldiers and were inferior to white soldiers. The result of this thinking meant that although colonial soldiers of African descent were allowed to serve in uniform, they were not allowed to fight on the Western front against white Germans and were instead given other roles working with different non-combat tasks. Outside of the Western front, however, the use of soldiers of color in combat roles were far more widespread. For pragmatic reasons and since the enemy was considered "less white" than western Europeans, soldiers from the British West Indies fought against Ottoman soldiers in the Middle East. The largest exception of them all were Indian soldiers that served on the Western front, the Middle East, and in Africa. The inclusion of ethnic minorities from British colonies as combat soldiers on the Western front during the First World War is therefore a long and complicated process which expressed itself in various and contradictory forms from one imperial army to the other.

These contradictions were not exclusive for Great Britain. Although Germany did not field soldiers from its African colonies on the Western front and accused France in its propaganda for unleashing the "primitive Africans" onto the white European race, there were Afro-Germans fighting for the Kaiser. France, unique out of all three main imperial powers on the Western front, did not hesitate to use West African soldiers whatsoever. Canada was hesitant at first but allowed their indigenous population to join and serve in positions where they'd see action against white men (while at the same time segregating black Canadians into the No. 2 Construction Battalion and giving them labour duties). Australia outright rejected Aboriginal men from serving in the Australian army in any and all capacities.