Born in Glasgow in 1973, Alyn grew up in Scotland and Saudi Arabia, returning to Scotland in 1986. Having studied Law and European Law, Alyn also gained a Masters degree in European Studies from the College of Europe in Warsaw. After a year teaching English in India and working with Scotland Europa in Brussels, Alyn qualified as a lawyer, practicing commercial law in London and Edinburgh.

In 2002, he started working for Richard Lochhead MSP, then the SNP Group at Holyrood as an adviser on European, justice and business policies. Alyn stood for election first in 2001 in Edinburgh West for Westminster, then again in 2003 for Holyrood, again for Edinburgh West. He was elected to the European Parliament for the first time in 2004 and has been re-elected three times in the SNP victories in the 2009, 2014 and 2019 elections, in 2019 as leading SNP candidate winning the best ever result for the SNP in a European election.

Alyn subsequently sought and won nomination for the Stirling constituency in order to fight for Scotland in Europe, and won Stirling in the December 2019 Westminster election with 51% of the vote. He pledged at the declaration to represent everyone in Stirling regardless of politics. In the Westminster Parliament, he is the SNP Europe and EU Accession Spokesperson.