Replace a field on a custom form with a new one and move the data from the old field to the new field. Can this be done without Fusion? If so, how can it be done?

Your options partly depend on the nature of the data in the existing field, and how many records already contain data in it. I'm also assuming that simply renaming your field's form display name, or re-labeling report column headers to clarify isn't an option here? 

When I've had minimal records that were previously completed, I added the new field to the form and saved. Then I built a report that showed both fields side-by-side, and copy/pasted while my coffee brewed. It's not often feasible but sometimes the easiest solution.

In some instances I've had way too much legacy data for that to work, but the legacy field was also a picklist/radio button style field. In those cases, I did the same as above but then cycled through each item in the legacy drop-down with report filters. Isolate all the records where <Legacy Field> = 'Red', and then made a mass-update on all resulting records to select whatever the corresponding value was in the new picklist. Rinse and repeat for each value in the legacy picklist.

In the really difficult ones, I've had legacy data that was either text or typeaheads where neither option above is practical. In those cases, then I've built a Fusion scenario to copy everything over. 

Can you create a report which isolates all of the old data and then using that report bulk move the values from one to the other by editing?

I use API calls to achieve this. If for example, the object type is Project, I will pull a project report in Workfront with (at minimum) the Project ID and the "Old field". I'll throw that in Excel, add a forth column, generate an API key for myself in Workfront, and write a HYPERLINK formula using the Project ID, "Old field", and API key values to then populate the new field. I still have to click in order to update the "New field", but it's less clicks than manually replacing the data within Workfront if I know I have a wide swatch of unique values.

For example, our instance unfortunately had two "Cost centre" fields that I had to consolidate. I pulled a Project report to display all projects where the "wrong" cost center field was populated. I then used the following formula in Excel:

You would replace MYINSTANCENAME and MYAPIKEY with your own, as well as the project ID ([@ID] in my table), new field ("DE:Cost%20centre" in my case), and old field ([Old cost centre field] in my table). Then, I would click the links which opens up a new browser window for each click, wherein you should receive a message that the information has been updated.

Not sure this is the simplest explanation, but hopefully it helps you or someone else reading!