He turned professional at Newcastle United in 2008, and went on to represent England at under-17 and under-18 levels. He spent time on loan at Gateshead in the 2011–12 season, and played eight Premier League games for Newcastle in the 2014–15 season. He joined Bradford City on loan in March 2015, and Port Vale on a free transfer that August. He was sold to Rangers for an undisclosed fee in January 2017.
Alnwick joined Newcastle United from local rivals Sunderland in 2008. He won the Wor Jackie Award for the club's best youth player in 2011.
He was loaned to Conference Premier club Gateshead in September 2011. After a man of the match performance in a 3–2 defeat to Southport at the Gateshead International Stadium on his first team debut his initial four week loan spell was extended until the end of the calendar year by manager Ian Bogie. He made a total of six league and three FA Cup appearances during his loan spell.
Alnwick signed a new two-year contract in July 2013. He made his debut for Newcastle United as a half-time substitute for the injured Rob Elliot in a 2–1 Premier League home win over Chelsea on 6 December 2014 and made five saves throughout the match. Two days later manager Alan Pardew declared that Alnwick was playing for a new contract with his current deal due to expire in the summer. A week later, in his first full start, Newcastle lost 4–1 to Arsenal at the Emirates Stadium. He made his first League Cup appearance in the quarter-finals at Tottenham Hotspur, and spilled a corner while under no pressure, which allowed Nabil Bentaleb to open the scoring in a 4–0 defeat. In his first start at St James' Park, Alnwick conceded in injury time from Sunderland's Adam Johnson as Newcastle lost the Tyne-Wear derby on 21 December. His final Premier League outing of the 2014–15 season came in a 3–3 draw with Burnley on 1 January. Tim Krul returned from injury for Newcastle's trip to Chelsea ten days later.
Alnwick joined League One club Bradford City on loan until the end of the 2014–15 season on 13 March 2015. The following day, he started away to Notts County in a 1–1 draw, but was cup-tied for their defeat by Reading in an FA Cup sixth round replay on 16 March. In Bradford's next league game, a 2–2 draw with Fleetwood Town at Valley Parade five days later, Alnwick was dropped to the bench as regular goalkeeper Ben Williams returned. On 24 March, Alnwick admitted that his Newcastle career may be over and regretted not getting loaned out earlier on in his career in order prove himself. His season ended in early April as he picked up a wrist injury which required surgery. On 29 May, Newcastle announced that they had opted against offering him a new contract.
After an unsuccessful trial at Championship side Leeds United, Alnwick joined League One club Port Vale on a five-month contract to cover for the injured Chris Neal on 6 August 2015. He kept a clean sheet on his debut, the opening game of the 2015–16 season, a 0–0 draw with rivals Crewe Alexandra at Gresty Road. He continued his highly impressive form, and was praised for a world class save to deny Sheffield United's Billy Sharp a goal during a 2–1 win at Vale Park on 3 October. He signed a new two-and-a-half-year contract in October 2015. He retained his first team place upon Neal's recovery from injury, and was named in the Football League's Team of the Week for "a string of fine saves" in a 2–1 victory over Barnsley on 14 November. His good form led to him being linked with moves to Wigan Athletic and Preston North End in the January transfer window, though he said that he was "over the moon" to stay with Port Vale. However a stomach injury he picked up in February caused him to miss four games, during which time Neal impressed in goal. He was returned to the starting line-up upon his recovery and went on to pick up the club's Young Player of the Year award, whilst Neal was released.
New manager Bruno Ribeiro appointed Alnwick as joint-captain, along with Ben Purkiss and Anthony Grant, in July 2016. He kept clean sheets in the two opening league games of the 2016–17 season, and was named in the EFL Team of the Week for his performance in the 2–0 win over Southend United on 13 August. He was named in the Team of the Week for a second time following his performance in the 1–0 win over Milton Keynes Dons at Stadium mk on 9 October. His third appearance in the Team of the Week came following his performance in the 0–0 draw with Oldham Athletic at Boundary Park on 2 January. Later in the month Rangers were reported to have activated Alnwick's £250,000 release clause.
On 30 January 2017, Alnwick signed a three-and-half-year contract with Scottish Premiership club Rangers for an undisclosed fee. Manager Mark Warburton signed him to provide competition for Wes Foderingham. He made his debut for the Ibrox club on the final day of the 2016–17 season, a 2–1 victory over St Johnstone at McDiarmid Park.
Alnwick was capped by England at under-17 and under-18 level and was named in Brian Eastick's squad for the 2011 FIFA U-20 World Cup in Colombia as back-up to Jack Butland and Lee Nicholls.
Alnwick was born in Hexham and is the younger brother of former Sunderland and England under-21 goalkeeper Ben Alnwick. The pair played against each other in a 1–1 draw between Port Vale and Peterborough United on 17 October 2015. Alnwick typically sports a large beard, though went clean shaven to raise money for the Movember Foundation in 2016.