Can I continue the desktop version of Home and Business on the mobile version ?

Unfortunately the desktop version saves your information to your local computer.  You won't be able to access information saved to your local computer from the online version.  If you used the desktop version last year you won't be able to access last year's tax return from the online version either.  

If you have someone who can access your computer for you they could email you last year's file and the data for this year's file that you have completed so far.  Then you could import those files into the online version or download the desktop version where you are at and continue on a different computer.  But the personal information for the desktop program is only stored locally on your personal computer.

Here is how to find the files on your computer.

@peter-soren