[4] In 2013, Twardek joined the academy of English Championship side Millwall and signed his first professional contract with the club in April 2015.

[4][5][6] After missing the start of the 2015–16 season, due to a dislocated toe sustained in July, he returned from injury on 8 October 2015, playing in the reserve match against Plymouth Argyle U21.

[9] Twardek made his senior debut for Millwall on 30 August 2016, in an EFL Trophy game against West Bromwich Albion under-23s, where he set up a goal for Jimmy Abdou, in a 2–0 win.

In the 2017–18 season, Twardek made his league debut for Millwall on 21 November 2017, coming on as a late substitute, in a 0–0 draw against Hull City.

[17] His performance at Braintree Town resulted in his loan spell being extended for another month and eventually for the rest of the season.

[18] He went on to make 12 appearances for the side, as Braintree Town were relegated from the National League at the end of the season after finishing in the bottom four.

[20] He made his Carlisle United debut, where he started and earned himself a Man of the Match, in a 1–0 win over Forest Green Rovers on 30 January 2018.

[25] Twardek made his Sligo Rovers debut, where he came on as a substitute for Caolan McAleer in the 69th minute, in a 2–1 loss against Bray Wanderers on 8 July 2018.

[35] In December 2021, it was announced that Twardek would return to League of Ireland Premier Division club Bohemians on a multi-year contract.

[37][38] On 30 January 2024, it was announced that Twardek had signed a two-year contract, with an option for a third year, with Canadian Premier League club Atlético Ottawa.

[41] On January 10, 2025, Canadian Premier League club Valour FC announced that Twardek had signed a one-year contract with them for 2025, with an option for 2026.

[6][44][45] In October 2013, Twardek was called up by Czech Republic national under-17 side, leading him to make his debut against Israel U17, in a 4–2 win.