Baldwin was on trial at Gillingham and impressed Gills boss Andy Hessenthaler after a season with Faversham Town's first-team at the age of 18 years before going on trial with North-East side Hartlepool United and signing a professional contract after impressing manager Mick Wadsworth in pre-season friendlies. Prior to signing this contract, Baldwin was due to start University in a few months time to do a course in Sports Therapy at Kent.
Baldwin made his senior debut for Hartlepool United on 17 December 2011 in a 1–0 home defeat against Colchester United with caretaker boss Micky Barron saying "Jack came in and was steady. He got on the ball, won his tackles and headers and competed really well. He's been terrific since he came to the club on trial in pre-season." Initially playing for the reserve side since the start of the season, Baldwin began to establish himself in the Hartlepool United's starting eleven, playing in either defensive-midfield and centre-back throughout the 2011–12 season. He went on to make seventeen appearances in his first season and shortly after the end of the 2011–12 season, Baldwin committed his future to Hartlepool by signing a long-term contract with the club.
In the 2012–13 season, Baldwin found himself in and out of the first team, where he was often featured on the substitute bench. After being suspended for one match and appearing three times as an unused substitute on the bench, Baldwin scored his first senior goal in a 2–1 defeat to A.F.C. Bournemouth in January 2013 and then scored again, four days later in a 3–1 win away to Portsmouth. Baldwin then set up a double for three goals, in a 3–0 win over Crewe Alexandra on 26 February 2013. However, during a 0–0 draw against Colchester United on 5 March 2013, he injured his right knee and was substituted before half-time. Initially out for the rest of the season, Baldwin returned to the training soon after and played his first match in weeks on 29 March 2013, in a 2–0 loss against MK Dons. At the end of the 2012–13 season, Baldwin went on to make thirty-five appearances and scoring twice times in all competitions.
In the 2013–14 season, Baldwin began to established himself in the first team once more at Hartlepool United, forming a partnership with Christian Burgess. He then scored his first goal of the season on 12 October 2013, in a 3–0 win over Exeter City. He went on to score two goals in the FA Cup Campaign against Notts County and Coventry City. On 29 December 2013, Baldwin scored again, in a 2–1 win over Morecambe. Following this, Baldwin was linked a move away from Hartlepool United, with League One side Peterborough United as among interests in him. Up until his departure from Hartlepool United, he was an ever-present player for the side, having started every league match since the start of the season.
Baldwin moved to League One side Peterborough United for £500,000 on 31 January 2014 just minutes before the transfer deadline. Upon joining the club, it was revealed the price tag could rise up to £750,000 with the potential for add-ons.
After being initially out of the first team, due to being suspended whilst at Hartlepool United, Baldwin made his Peterborough United debut on 1 March 2014 against Crawley Town, coming on as a late second half substitute, and conceded a penalty in a 1–0 loss. Since making his debut, Baldwin quickly became a Peteborough's first team regular and helped the side reach the play-offs where they were unsuccessful after losing 3–2 to Leyton Orient. His performance was praised by Manager Darren Ferguson. At the end of the 2013–14 season, Baldwin went on to make thirteen appearances in all competitions.
The 2014–15 season, however, saw Baldwin suffered setback despite being featured in the first team regularly After playing in the opening game of the season against Rochdale, Baldwin, however, suffered a knee injury that kept him out for the remainder of August. Two months later, on 21 October 2014, Baldwin was then sent-off for a second bookable offence, in a 1–0 loss against Crewe Alexandra. On 1 November 2014 Baldwin captained Peterborough for the first time. In the same game he ruptured a knee ligament that would rule him out for the next ten to twelve months. Despite this, Baldwin made his full season, making eleven appearance in all competitions.
The 2015–16 season saw Baldwin continuing to rehabilitate his knee injury and it wasn't until on 22 September 2015 when he returned from injury, playing for Peterborough United's development side, in a 3–1 win over Norwich City's development side. It wasn't until in October when he didn't return to training. After returning to the first team on the substitute bench against Scunthorpe United on 28 November 2015, he made his first appearance of the season on 26 December 2015, coming on as a second half substitute, in a 2–0 win over Chesterfield. Although sidelined on two occasions in February 2016, Baldwin played his first match as captain on 5 March 2016, in a 1–1 draw against Wigan Athletic and went on to captain on four occasions. Baldwin then scored his first goal for the club on 30 April 2016, in a 4–3 win over Shrewsbury Town. Having established himself in the starting eleven with Ricardo Santos, Baldwin went on to make twenty-two appearances and scoring once in all competitions. For his performance, he was awarded PFA Community Champion by the club.
The start of the 2016–17 season saw Baldwin being booked for the first three league matches before being sent-off for a second bookable offence, in a 2–1 loss against Oxford United on 20 August 2016. After serving a one match suspension, Since the start of the season, Baldwin featured in every league matches until he was demoted to the substitute bench for two matches due to good performances from Ryan Tafazolli. However, he suffered a knee injury that kept him out for two months. It wasn't until on 21 February 2017 when Baldwin returned to the first team and captained the side as well, in a 4–1 loss against Southend United. Four days later, on 25 February 2017, Baldwin captained and scored again, in a 3–1 win over Rochdale. Despite being absent on two occasions later in the season,, Baldwin finished the 2016–17 season, making twenty-nine appearances and scoring once in all competitions. On 25 May 2017, Baldwin signed a new two-year contract with the League One side.
Ahead of the 2017–18 season, Baldwin was named the new captain of Peterborough United, succeeding Christopher Forrester.
Baldwin is a versatile footballer who can play centre-back, right-back or central midfield. Former Hartlepool manager Neale Cooper likened 18-year-old Baldwin to a 'young Alan Hansen' praising Baldwin on 'the way he reads the game'.As of match played 16 September 2017
Up until joining Hartlepool United, Baldwin attended Eastbury Comprehensive School and was at the time studying for his A-level exams.
In 2014, Baldwin became a father and cited family reason as the reason he left Hartlepool United. Later in 2014, Baldwin took part in a cycling challenge with Andy Rafferty for the 120-mile ride from Whitehaven, in Cumbria, to Blackhall in aid of raising money for London-based Richard House Children's Hospice.