One suggestion I have so far is a permissions/authentication issue.
On your server, go to IIS Manager
Then open the the tree > Server name > Sites > Default Website > ArcGIS
Then click on Services, then double click on Authentication.
You can adjust the authentication type, try Anonymous = Enabled and all others set to Disabled.
Or try the other way around and have Windows = Enabled and all others set to Disabled.
I am still using ArcGIS 10.0 so I am using an older version of ArcGIS Mobile as well.
You can also code your own synchronization method, have a look through the forums, I believe someone or I have posted the code on how to do it. In addition, when you code your own sync, you can grab the error.
For example, when I change the authentication I get 401 Unauthorized.
I am curious as to what error number and error you are getting.
Synchronization is a HTTP request and your IT/Network admin can help troubleshoot it.
I hope this works or sheds some light on this.
Another thought would be to install your application on your desktop which already has an Ethernet cable plugged in and attempt a sync. If that works that is great.
Just and FYI, I am also using cell SIM cards and Internet sticks for my tablets and laptops, but I am also using a VPN connection to gain access to the network. Meaning my field computers/tablets appear to be physically connected to the network because of the VPN. Whereas if I did not have the VPN connection my field teams would not be able to sync. I'm not sure what the error would be but my guess would be 404 Unable to get access.