How much NM cable can I fit in a conduit?
For conduit fill calculations you have to treat romex/NM as a ROUND wire of the same size as its largest dimension (ie, the long axis of the oval.) Oddly enough this means that 14-2 is larger than 14-3 for conduit fill. As @SpeedyPetey notes nipple (less than 24") fill is 60%, not 40%, and is a LOT of wire to get into a conduit; but I don't think that's the major issue here if NEC compliance is a desired upgrade. The wire type and location is the main issue.

NM cannot be run in exterior/underground conduit since it is not rated for wet locations. With an outside service box that alone might mean you can't do this and meet code - at which point you'll need a major junction box or a lot of minor junction boxes inside, and some wet-location-rated wire until you are inside.

(And now that I've seen the pictures...)

My understanding of NEC (which does not seem to have been particularly involved in your current service, with exposed NM running outside) is that you would have to run a wet-rated wire (THW, XHHW, etc.) until you get inside the house, where you would connect to the interior type wire you currently have exposed to weather and sunlight. NEC defines (and reality frequently confirms) all exterior conduit as a wet location.