Former Rochdale and Tranmere midfielder Thompson has joined on a one-year contract, with the inspirational 25-year-old having battled back from cancer over the past 12 months.
"Bury have given me the opportunity to rebuild my life and get my career back on track," declared Thompson on Twitter. "For that, I am truly grateful."
He will be joined in David Flitcroft's squad for Saturday's League Two opener at home to Cheltenham by Widdowson, who played under the same manager at Rochdale and has put pen to paper on a short-term deal.
The 25-year-old midfielder came through the ranks at West Ham and has also represented Rotherham, Grimsby and Northampton, playing 28 times for the Cobblers last term.
However, James Burke will move on by mutual consent after moving to the JD Stadium from Huddersfield in January but failing to secure a regular first-team place.