Discussing the speech on Newsnight, radio host and right-wing commentator Julia Hartley-Brewer said it ticked every box and Leave voters had "got their cake" and were eating it too.
But Polly Mackenzie, director of policy institute Money and Mental Health, said the speech was difficult to swallow.
On Tuesday, Mrs May gave the clearest indication yet of the type of Brexit she will pursue – announcing Britain would leave the single market and most likely the customs union too.
Ms Mackenzie said: “I think everybody who voted Remain is basically grieving today.
Julia Hartley-Brewer said most Leavers would accept no EU deal as a worst case scenario
I think everybody who voted remain is basically grieving today.
"People have held on for six months to this idea that there might be some kind of soft Brexit, we might be able to have a Norwegian model or a Swiss model or something.
"Theresa May then made the decision, it's about immigration at any cost and that means we can’t be part of the single market, we can’t be part of the customs union.”
Asked by presenter Evan Davis whether the country can unify and move forward, Ms Mackenzie added: “It’s a bit galling to be told to unify with somebody you fundamentally disagree with and just get in line because sorry you lost.”
An irate Hartley-Brewer then interjected: “What are the other options? You lost. So we should just say, we won’t leave the EU, even though more people wanted to leave the EU than wanted to stay, in a binary vote.
"I don’t understand what compromise you think there is.”
She added ‘soft' Brexit wasn’t Brexit but Ms Mackenzie said a lot of people had been “clinging to that hope” the UK would retain at least partial membership of the customs union.
Ms Mackenzie, a former policy director to Nick Clegg during his time as Deputy Prime Minister, said: “I think when you look at the polling there are a big group of people who say that they would prioritise single market access and economic access over immigration.
Theresa May's 12 point Brexit plan
Mon, January 16, 2017
It's finally here!
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Continued cooperation with the EU to tackle terrorism and international crime.
"It is the people who, like Theresa May, have decided immigration comes first, which a substantial part of the population – probably 40 per cent – who are going to be dancing on the table tops today."
Hartley-Brewer retorted: “Most of the Remainers I know, and the polling shows very clearly, many of them just want to get on with it and they’re happy and they accept the democratic vote.”
She added she accepted the UK could end up with no deal and she “like most Leavers” was willing to accept that in the worst case scenario.