AWS Migration Service not synchronising changes from Source server

You're right about data continuing to be replicated until you have completed the final cutover. The data is always replicated only to the EBS volumes attached to the replication instances and the snapshots that MGN continuously makes of them. The test instance you launched is a snapshot of the server as it was just moments before the test instance was launched (specifically, the moment when the EBS snapshots were started). MGN won't modify data on the test instance after it's been launched.

To see changes made on the source server after the test instance was launched, you'll have to complete or cancel the current test and launch a new test instance.

It would be practically infeasible to keep a running server replicated with changes from a source server in a meaningful way, simply due to the ways that operating systems are designed to rely on their autonomy in managing their memory, file systems, and the underlying block storage. The upside is that this structure also allows you test your post-cutover activities, such as removing VMware Tools, hardware drivers, old monitoring or backups tools, and so on, in a way that matches the final cutover. You can rehearse the post-migration steps as many times as you need to get it to go just as you want by cancelling/completing the test and starting a new one.