Bashir, who is of Pakistani heritage, returned to London from Abu Dhabi, where England trained before arriving in India, and will now join the squad this weekend after his visa issue was resolved on Wednesday.
"Shoaib Bashir has now received his visa, and is due to travel to join up with the team in India this weekend," England's cricket board (ECB) said on social media platform X.
"We're glad the situation has now been resolved."
Earlier, Stokes did not hide his disappointment in his pre-match press conference but said they never seriously considered boycotting the match over the treatment meted out to the Somerset spinner.
"I'm pretty devastated that Bash has had to go through this," Stokes said.
"As a leader, as a captain, when one of your teammates is affected by something like that you do get a bit emotional.
"I know he's back in London and a lot of people are jumping through hoops to try and get this through quicker.
"Hopefully we're going to see him here over the weekend."
India and Pakistan have soured political relations, and they do not play each other outside multi-team events.
People of Pakistani origin are required to submit additional details for obtaining an Indian visa, according to the website of the Indian High Commission in London.
Australia's Pakistan-born opener Usman Khawaja faced a similar visa delay ahead of their tour of India last year.
The Pakistan Cricket Board also complained about a visa issue ahead of last year's 50-overs World Cup in India.
Lancashire's Tom Hartley is set to make his test debut in Hyderabad after Stokes announced a spin-heavy attack also including fellow left-armer Jack Leach and teenage leg-spinner Rehan Ahmed with Joe Root as the fourth option.
James Anderson has been left out with Mark Wood as the only frontline fast bowler for what England expect to be a spin-dominated contest.
"It would be a complete and utter 'gut' thing, as most of my decisions like that are," Stokes said of their team selection.
"If (left-handed) Yashasvi Jaiswal opens the batting (for India), you might see Rooty opening the bowling because it's spinning away from the bat."
Ben Foakes returned as wicketkeeper, freeing Jonny Bairstow to play as a specialist batter at number five.
