Image copyright EPA

The prime minister seemed pleased with the outcome of her trade visit to India. Theresa May said the 33-strong delegation had sealed deals worth £1bn, laying the groundwork for more than 1,300 jobs in the UK.

Mrs May even donned a sari on a visit to a Hindu temple in the southern city of Bangalore, mixing the traditional with the modern. Bangalore has now become an important IT base for many other countries.

Yet in India, they're not so sure. Her reception was polite, with Prime Minister Narendra Modi granting her an unusually long meeting of 90 minutes, but there are now significant tensions in the relationship which three days of photo opportunities will not resolve.

Lord Bilimoria, the Indian-born businessman who created Cobra beer, has been on many of these trade trips stretching back 20 years, with both Conservative and Labour prime ministers.

Image copyright Getty Images Image caption Theresa May and her Indian counterpart Narendra Modi met for 90 minutes during her visit to Delhi

On The World This Weekend on Radio 4, he told me that Indian companies had invested more in the UK than anywhere else in Europe. They've seen it as the bridge to the EU. As a result of Brexit, that bridge is about to be broken, so having good links with the UK will become less important.

That means India could, in future, see Britain as a less essential trade partner, just as the UK needs its enormous market even more.

The problem is exacerbated by British decisions about immigration. The treatment of Indians who have studied in the UK and want to remain, and those who come as tourists or to work, rankles the Indian government. It's Mrs May herself who's blamed.

At the Home Office, she presided over an increasingly restrictive policy, in contrast with the Cameron government's treatment of wealthy Chinese, who have seen many obstacles to their entry removed.

Leaving Europe could improve the situation. As home secretary, Mrs May cut back on non-EU migration, to try to meet the challenging target of reducing net migration to tens of thousands of people.

She's powerless to cut the numbers of citizens from other EU countries coming to the UK. Post-Brexit, she could spread the reduction more evenly.

Image copyright Stuart Wilson/Getty Images Image caption Lord Bilimoria said that, when home secretary, Theresa May delivered "very negative messages towards immigration"

As president of the UK Council for International Student Affairs, Lord Bilimoria told me that a particular source of tension is over the treatment of students.

"The reality is Theresa May - when she was home secretary - did deliver very, very negative messages towards immigration," he told me.

"We continue to include international students within our net migration figures," he added.

"The perception that sends out to the rest of the world, including India, is Britain wants to reduce the number of international students."

In that, the policy has been successful, with numbers falling from 40,000 to 20,000 in the last five years. Successful in British terms, perhaps, but Lord Bilmoria says, not to Indians.

"Theresa May as home secretary is the one who said that: 'I want international students to leave the day they graduate.' And the headlines in India at that time were: 'Take our money and then get out.'"

In our interview, Lord Bilimoria revealed to me that last year David Cameron, then the prime minister, had suggested policy was about to change.

"David Cameron… told me personally in November: 'Karan, I think we should take international students out of the net migration figures.' It didn't happen."

Whether he changed his mind, or lost a battle within Cabinet, the Indian government was very disappointed that no such announcement ever came.

Nor did it this time. Theresa May did propose speeding up the application process for India's businesspeople, but said other changes would only be considered if the deportation of Indians who had overstayed their welcome in the UK could be speeded up.

Image copyright Getty Images Image caption India's Commerce and Industry Minister Nirmala Sitharaman says the UK no longer treats India as an old friend

As for students, she said they were welcome, and pointed out that nine out of 10 student applicants did receive a visa. That may be true, but are there as many applications as there were in the past?

Currency questions have been distracting Indians in the days since her visit, after the dramatic decision to cancel the 500 and 1,000 rupee notes. So the average man or woman in India may not yet have fully absorbed Mrs May's visa resistance.

There are reasons, then, to doubt how quickly India will want to reach a trade deal when Brexit happens.

Earlier this year, Foreign Secretary Boris Johnson was optimistic that Britain's partners in the world outside the EU would jump at the chance, observing: "We used to run the biggest empire the world has ever seen, and with a much smaller domestic population and a relatively tiny civil service.

"Are we really unable to do trade deals?"

I quoted that to the Commerce and Industry Minister Nirmala Sitharaman, suggesting that India would therefore be better disposed towards Britain than to some other countries.

"Well, I thought so and most of us in India thought so; but we aren't being treated as old friends any longer. It's a tight professional engagement," she said.

In such relationships, calls to old loyalties or sentiment don't count for very much.

You can hear The World This Weekend on Sunday lunchtimes at 13:00 GMT on BBC Radio 4, and via the BBC iPlayer.