I have also experinced these problems on the C2 and the AAS where the connectivity through mobile data sometimes breaks and is at times painful to get back.
However, I think I understood what the problem might be and how to recover if it happens.
I think the AAS cannot keep up with the native layer if you change between WiFi and mobile data or between different providers (SIMs) for mobile data.
For me, with 2 SIM cards those 4 steps by @hschmitt were not enough to get mobile data for AAS to work after it was lost.
But I found a simple fix in the end (no need to restart AAS and disable and re-enable SIMs):
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Go to the AAS settings and tap the verion number 5 times to get to further details/settings.
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While there, go to the Network & Internet settings and make sure you have the same (if using multiple SIMs) operator’s connection selected as on the native Sailfish side’s settings.
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Then go to the SIM-card settings and more specifilally to that SIM card’s settings that you are using on the native side and make sure that mobile data is enabled (mine wasn’t) there.
After this I got the mobile data connectivity for AAS back online.
So, I think these problems could be avoided if the native settings would “command” the corresponding AAS settings more closely 