I offer two free tools that you could use for this purpose:
- With Upgradehub you can go from 1.4.9 until 1.6.1.24
- With Copy_shopdata you can go from 1.5.0.11 up to the latest version.
Upgradehub uses a lightly modified version of the Prestashop upgrade scripts but uses them in such a way that you see the errors, can fix them and then continue where you were. It requires some study but then you are almost guaranteed that your upgrade will succeed.
Copy_shopdata copies the business data to the database structure of the new version, making fixes where needed.
If you want use the Prestashop methods it is best to use the manual upgrade method but I am not sure whether that works across the 1.4/1.5 barrier.
The 1-click upgrade module is less recommended: it wants to download its latest version. But when you do that you will see that it cannot run with your PHP version. There is a workaround for that by changing the version number in its config file so that it believes that it is up to date.
Finally there are the migration modules/services like Cart2Cart and MigrationPro. Just like Copy_shopdata they copy the business data. The difference is that they tend to be general tools that also support Woocommerce, Opencart, etc. So they may miss some of the finer details. Usually that is not a problem, but you should do some checks after the migration. Prestashop's Addons has a section for migration tools.