Safe change of field type from text to email or link
A different approach  could be - Why try to change field type ? Why not copy old to new ?

You could possibly use something like views data export or other similar module to create a dump of your nodes as node ID and text field to csv file.

Then create your new field.

Then use feeds to import your csv and assign the old text field column to your new field. Make sure you tick the box to update nodes !

Then delete your old field.