There were at least several other women trained as cosmonauts. Tereshkova got the nod for political reasons similar to Yuri Gagarin--like him, she came from distinctly humble, peasant/proletarian origins, and was thus deemed the best choice politically. Unfortunately, unlike her male counterpart who was an experienced airman, she had had relatively little experience flying. Other female cosmonaut candidates (kosmonauta?) had had a lot more flight time, but the reason they were able to was that their class origins were not so iconically proletarian--being daughters of relatively well-off Russians, they were better positioned to get the test pilot gigs and so forth--and that class "taint" put them out of the running for first Soviet woman in space. But Tereshkova had a hard time handling zero gravity and the misfortunes of "Chaika" ("Seagull," her mission call sign) put further women in space far down the priority queue and under the circumstances, political and budgetary, they never got around to it until a later generation when the Americans were finally going to send up a woman of our own; a female cosmonaut was hastily added to a Soyuz mission to pre-empt Sally Ride for the title of second woman in space.
So, either Tereshkova having more flight experience and better luck in adapting to zero-g, or her being scrubbed from the first mission and one of the other kosmonauta being chosen and performing better, might have left the door open for yet more and the normalizing of something like gender equity in the Soviet program.
(Sort of. Decades later in the Salyut and early Mir missions, male cosmonauts would still present arriving women with the traditional Russian gifts to welcome a woman to a house--an apron and a broom!

The infamous "second shift" for women lived on In Space...)
As for what this would have done for the American program--well, it was a lot more odd for the Russians to omit women from their space program than for Americans, since the Soviets were supposed to be progressive Communists whereas Americans were supposed to be more conservative. But actually there were arguments--largely ignored--for American women in space from before the Mercury program launched anyone. How this would have played out in the 1960s, and in the Women's Lib era of the early '70s, is the stuff of actual timelines of course!