Farmers, health chiefs and industry bosses have all expressed concern that the system will make it much more difficult for immigrants to qualify to enter the UK.
Agriculture in Boston and the rest of Lincolnshire could be particularly badly hit, as ‘overseas’ people form the majority of the workforce.
However, Ms Atkins, a Home Office minister, says she does not believe there will be any shortages - but adds the Government is continually monitoring the situation.
The Louth and Horncastle MP said: “In the 2016 referendum on whether or not to stay in the EU, many areas of Lincolnshire had some of the largest percentages of leave votes in the entire country.
“Going around my own constituency and talking to people, the main reason for people voting to leave was a real desire to take back control of our borders - and reduce the number of overseas’ workers.
“We have done exactly that with this new system.
“If people are now complaining, my message to them is simple - ‘you can’t have it both ways’.
“You asked for more controls and that is exactly what we are delivering”
Ms Atkins stressed the legislation did identify key roles in which the qualifying system for points would be ‘relaxed’.
She said this would apply to, for example, the medical profession, enabling recruitment of overseas’ nurses
and doctors to continue.
Ms Atkins went on to point out most of agricultural workers were ‘already in this country’ and had applied to stay on after Brexit.
She added: “There are lot of headlines about the potential problems but we need to put things into perspective.
“And, if we want to control our borders, then we have to have some form of regulation in place.
“I am aware how many overseas workers are employed in agriculture (in Lincolnshire).
“They perform a very important role and make a vital contribution to local economies.
“No-one is telling these people they have to leave. We are monitoring the situation and there is still time to make changes to the system - if any are needed.”
Ms Atkins pointed out the Government has provided extra funding to recruit thousands of new nurses.
She believes many overseas nurses, doctors and dentists will continue to want to come to the UK to work.
Locally, several landowners have raised doubts about whether there will be enough seasonal workers to pick crops in fields - adding ‘British people don’t want jobs’ which often involve early starts - and low pay.