When you’re onboarding software developers, which stage of your checklist or process do you think makes the biggest impact on their initial engagement and productivity?

The time taken for the organisation to give the new colleague the correct and working environment, tools, access and supporting process knowledge to get the point they can commit code for first time.... 

This is a timely question as we are in the process of onboarding full-time developers out of our internship program. 
1- Internships: I'd say that it is best to measure a candidate's engagement and productivity during an internship program as they are working on tangible projects which are objective in this regard.
2- GitHub / Open Source: For direct hires, I always check a candidate's GitHub page to see what projects they have posted, if any. It is a very positive signal for a candidate to have open-source contribution experience as it typically, albeit indirectly, signals a high level of engagement and productivity.
3- Technical Interviews: If a candidate is able to properly communicate during technical interview questions, it is clear that they have dedicated significant time and resources toward learning their craft, and this innately requires immense levels of dedication.

Onboarding in a large, heavily regulated organization, is subject to a potential series of rolling failures across a very complex process, especially for technologists who need tools and highly-scrutinized access. There's no particular step, rather it's more important that people don't get bogged down, that they can get through all of the steps as efficiently as possible. The biggest success factor is to have a goal of making a difference in the first few days, e.g. configuring a firewall, getting a first code commit into production, completing a workflow, etc. Define that goal based on the responsibilities of the team and then, as management, support the peers of new joiners in removing obstacles to shorten that time down to a couple of days at most. Create a wiki page called "I'm new here", encourage new joiners to update that page as they hit problems and solve them.

agree with previous comments, where onboarding with immediate access to required tools and information is essential. but it is a continues journey where assigning a right 'buddy' will ensure continues engagement.

During onboarding, the "environment setup and access" phase has the biggest influence on a software developer's early engagement and productivity. Making sure they have immediate access to the required hardware, software, and systems allows them to begin contributing right away.