Oswell, 24, who scored 22 goals in 2016-17, started his career at Crewe and joined Newtown from Airbus UK in 2014.
"It is disappointing to lose Jason, he has been a fantastic player for us in recent years," Newtown boss Chris Hughes said.
"But he expressed desire to return to the English system."

Michael McFeat from Perthshire was expelled by the Kyrgyz authorities after comparing the national dish with the private parts of a horse.
Speaking to the Sunday Post newspaper, Mr McFeat described how he was helped to escape "a lynch mob".
He has now been banned from Kyrgyzstan for five years.
Mr McFeat told the newspaper he actually believed the traditional Kyrgyz dish - chuchuk sausage - was a horse's penis, but the remark on Facebook angered his local colleagues.
He had been working at the Kumtor gold mine, but had to be smuggled away after the incident.
When he arrived at Manas Airport in Bishkek, he was arrested by police under race hatred laws.
Mr McFeat said: "The police told me my act could send Kyrgyzstan to war with the UK."
After appearing in court and making an apology for his remarks, Mr McFeat was put on a flight to Edinburgh.
He added: "I was told there was a 17-page petition demanding I be jailed and the mine went on strike after I left, so they were making an example of me.
"I've always been up for a joke but this was one time I wasn't joking and it's been blown out of proportion."

The Markit/CIPS construction purchasing managers' index rose to 56 from 53.1, its highest level since December 2015. A figure above 50 indicates expansion.
Builders also hired more workers and ordered more supplies to deal with an increase in new projects.
Economists said the sector was recovering after a slow start to 2017.
"A sustained rebound in residential building provides an encouraging sign that the recent a soft patch for property values has not deterred new housing supply," said Tim Moore, senior economist at IHS Markit.
"Instead, strong labour market conditions, resilient demand and ultra-low mortgage rates appear to have helped boost work on residential development projects."
The data contrasts with reports earlier this week suggesting greater caution by homebuyers and lenders.
Building society Nationwide said house prices had fallen for a third consecutive in May - the longest run of decline since 2009, just after the financial crisis.
Meanwhile, the Bank of England said loan approvals fell to a seven-month low in April.
The latest Markit/CIPS construction reading was still much weaker than the post-crisis high of 64.6 seen in January 2014.
However, the survey found there had been solid rises across the whole sector in May, including in civil engineering and commercial building.
Input prices for construction firms also rose at the slowest pace for seven months, despite the prolonged weakness of the pound.
Samuel Tombs, chief UK economist at Pantheon Macroeconomics, said the result suggested the Brexit vote's "dampening influence on construction activity is fading".
But he added: "Note, however, that the PMI has had to exceed 53 in the past to signal growth.
"May's PMI reading, therefore, is consistent with quarter-on-quarter growth in construction output of only about 0.5%."

Ms Leigh won an Oscar for her portrayal of Scarlett O'Hara in the 1939 movie.
Heritage Auctions said the grey jacket and skirt worn by the actress were among 150 items offered by collector James Tumblin.
He originally bought the outfit for $20 from a costume shop.
Welsh Premier League top scorer Jason Oswell has left Newtown for National League North side Stockport County.
[NEXT_CONCEPT]
A Scottish worker deported from Kyrgyzstan has said police there warned him a joke about a traditional sausage could have led to war with the UK.
[NEXT_CONCEPT]
Activity in the UK construction sector expanded at its fastest rate in 17 months in May, boosted by a surprise rebound in homebuilding, a survey says.
[NEXT_CONCEPT]
An outfit worn by Vivien Leigh in the film Gone with the Wind has been sold for $137,000 (£91,000) at an auction in California.