It is fair to say that free markets played a non-trivial role in industrialization and that protectionism in Britain hindered industrialization. However, the Industrial Revolution was caused by many other factors than having relatively free markets. It required Britain to generally adopt set of inclusive institution (of which relatively free markets are just one example), and it also required some historic luck.
In economics, it is accepted that countries with good 'inclusive' institutions, such as strong property rights, are more productive and able to develop faster (or even develop at all) than countries with bad 'extractive' institutions, such as forced labor (see Acemoglu 2008, Acemoglu & Robinson 2000a, 2000b, 2001, 2006, 2008; Olson 1984, Bates 1981, 1983, 1989 and sources cited therein)
An industrialization, is just a period of a rapid economic development where the share of national income from manufacturing also increases (Bagchi 2016), so it requires that country predominantly accept inclusive institutions which are thought to be the main precondition for an economic development.
Inclusive institutions are institutions that allow people broadly to participate in the economy, so free markets and free trade are generally inclusive institutions. However, they are not the only inclusive institutions and not even the most important ones. For example, a far more important institutions are strong and secure property rights, without which people would be afraid of making large capital investments (see Acemoglu & Robinson Why Nations Fail) which are obviously pre-requisite for industrialization to naturally occur. Next, having a central government that can impose peace and security from violence and coercion is also extremely important, forced labor or other types of coercive extractive institutions are antithesis to economic development (again see Acemoglu & Robinson).
Protectionism, is also an extractive institution, and Britain indeed was protectionist during its main industrialization phase, but in the grand scheme of things having few extractive institution, while relaying mostly on inclusive institutions won't hinder economic development. But generally it is argued and it was already recognized by contemporary economists that the Britain's protectionist policies (e.g. corn laws) hindered the industrialization and that it happened in spite of them rather than thanks to them, since its protectionist policies primarily advantaged agriculture (see Hirst, (1925) From Adam Smith to Philip Snowden).
In addition, it was likely also due to some historical luck. Other countries at that time, such as the Netherlands, had by all accounts more advanced economic system to that of Great Britain (see Mokyr 2000). In fact, while industrialization proper, started in the Britain, the Netherlands was already experiencing proto-industrialization at small scale far earlier (Bavel 2003). However, in the end the industrialization first took of in England, and it is argued that was because the Industrial Revolution at its inception requires a close cooperation between scientists, engineers and businessmen (see Mokyr 2000) . Mokyr argues it was the "closeness of natural philosophers, engineers, and entrepreneurs was a key to success in Britain."
Consequently, it would be fair to say that free markets played an important role in Britain's industrialization, but they were certainly not the only cause, far from it. Industrialization required a system built predominantly of inclusive institutions where private property, security, fair set of laws (at least relatively fair for their time) etc dominate. In addition, there already were countries that were undergoing proto-industrialization for much longer time than UK, but those countries did not undergo proper industrialization because share of income generated from manufacturing was still small. This was because for manufacturing to be really productive you need to apply some technology to it and UK happened to be a place where cooperation between scientists, engineers and business people was more common. Hence you should not just try to pin the industrialization in the UK on one single factor.