Prime Minister David Cameron's government last month announced plans to reopen the British embassy in Tehran, less than three years after the outpost was shuttered in response to a November 2011 mob attack by pro-regime students. In the interval, London had backed tough sanctions on Iran's nuclear program, condemned the Islamic Republic's role in the Syrian slaughter, and decried the mullahs' human-rights abuses at home.

But now British diplomats strike different notes. "There has never been any doubt in my mind that we should have an Embassy in Tehran if the circumstances allowed," Foreign Secretary William Hague said in a June 17 statement.

So what has changed? The thaw is in part the result of the global charm offensive launched by Iranian President Hasan Rouhani and his foreign minister, Javad Zarif, since they took office last year.

Take the Rouhani government's sophisticated diplomacy over the 2011 embassy attack. "An apology is not under consideration," Iran's deputy foreign minister for European and American affairs said in May. "But we are definitely working with the British government on what happened." The discussions might include compensation for the physical damage to the embassy compound, according to Iran's state-run PressTV.

The current rapprochement was also aided by a bipartisan trio of influential lawmakers. They include Jack Straw, who served as foreign secretary under Tony Blair ; Ben Wallace, a Conservative MP who with Mr. Straw was co-chairman of the All-Parliamentary Group on Iran; and Lord Norman Lamont of Lerwick, a Tory peer and former chancellor in the John Major government who heads the British Iranian Chamber of Commerce.

Messrs. Straw, Wallace and Lamont have in recent months criticized the obstacles posed by American sanctions to U.K. banks that do business with Iran. "The impact of this unilateral, extraterritorial jurisdiction of the U.S. is especially discriminatory against U.K.-based financial institutions, because of their multinational nature," Mr. Straw declared in a March Westminster debate.

I spoke separately with all three lawmakers on the phone last week. All three see engaging Tehran as good for British business and essential to stabilizing a Middle East tumbling toward chaos. Dissolved in this "realist" tincture is a measure of postcolonial guilt and anger toward Washington hawks who have shown "gratuitous hostility" toward Iran, as Mr. Straw put it to me.

All three sounded fairly clear-eyed about the realities of the Iranian regime. Nonetheless, their views on the conflict between the West and the Islamic Republic at times strikingly echoed Iranian talking points.

For example, asked if London should demand an apology in addition to financial restitution for the 2011 embassy assault, Mr. Wallace responded that "they've expressed regret" and cited the Western-backed 1953 coup against the nationalist Iranian Prime Minister Mohammad Mossadegh as a legitimate Iranian grievance.

Asked about the chants of "Death to America" and "Death to Britain," which resound weekly at Friday prayers in Mr. Rouhani's Iran, Mr. Straw said: "Do I approve such rhetoric? No, I don't. It's not a reason to not try and build a relationship. . . . Do I understand why they say it? Yes, because you have to understand Iran's history. . . . They remember and will recite the overthrow of Mossadegh in '53 and other humiliations by the U.K. or the U.S. or both in great detail."

Iranian officials are fond of using the coup as a moral alibi for regime misbehavior abroad. But they conveniently elide the fact that Islamist clerics, whose revolution they inherited, also supported the anti-Mossadegh coup.

Then there is the impression, also nurtured by the regime, that the Iranian leadership is divided between "moderates" and "hard-liners" with genuine ideological differences rather than tactical disagreements.

"I think the mere appearance of President Rouhani obviously encouraged business confidence," Lord Lamont told me. "The ministers whom we met were quite impressive, I have to say, all with American degrees. . . . The question is: Do these ministers really represent the regime? And that's where it gets difficult. We all know that the president of Iran has limited power. But I believe he's been given the go-ahead to probe with these negotiations."

Tehran can "use its power for stability and peace" and "earn a place at the table" as a result, Mr. Wallace told me. Asked if the mullahs have earned such a place thus far, Mr. Wallace responded: "Well, has America, after the invasion of Iraq, earned the right to play a role in the Middle East—or Britain?"

Along with Jeremy Corbyn, an anti-imperialist firebrand and Labour MP, the three traveled to the Islamic Republic in January. On their visit, the British lawmakers sat down with Mr. Zarif, Mr. Rouhani's chief of staff, trade officials and various parliamentarians. They didn't attempt to meet any dissidents or civil-society activists inside the country, they said, though Mr. Wallace says they raised human-rights issues with Iranian officials.

Mr. Straw has since painted an optimistic picture of what his delegation saw and heard in the Iranian capital. "Tehran looks and feels these days more like Madrid or Athens than it does, say, Mumbai or Cairo," he wrote in a January op-ed.

"I know that Tehran is not Madrid," Mr. Straw told me. "My point was that's what the city felt like from the narrow perspective of the journeys that I was making. That was all. It feels more like those cities, Athens too, it felt to my entirely subjective judgment, than, I think, Cairo or Mumbai—what it felt like looking out from the car."