Installing a brake cable may cause shifting problems because brake cables (and their hosings), in addition to being thicker (hosing is usually 5mm vs 4mm on gear hosing, cable is usially 1.5--1.6mm vs 1.1--1.2mm), are slightly flexible---cables stretch a bit more, and especially hosings compress a bit more. It is because with braking, they need to bear more weight but precision is less important (you basically adjust braking power using the feel in your hand and by how the bike feels). A couple mm difference can occur at the length of some 1--1.5 m and you would feel it.
That being said, if all three lowest gears are not working, this is not likely to be a problem. It is because lowest gears on Shimano systems are the ones where the cable has the most slack, i.e., lowest tension. So assuming the cable moves freely in the hosing, the problem is not the cable.
To diagnose, take cable nut (what Shimano manuals e.g. https://si.shimano.com/en/pdfs/sm/IHG-INTER7/SM-IHG-INTER7-003-ENG.pdf call "inner cable mounting bolt") off from the cassette joint, and take the hose off as well. See if the length appears correct (you can measure it using a ruler - see dimensions in section "Installing to the cassette joint" of the manual I linked above), and see if the cable moves all the way back and forth as you switch between lowest and highest gear on the shifter handle.
If that looks okay, the problem is either with the cassette joint (e.g., it is broken or stuck) or in the gear hub itself. For starters, take the wheel off, take the cassette joint off, clean up everything that seems dirty, reassemble, try again.
If this does not help, and if the cassette joint (Shimano part CJ-NX10 or CJ-NX40) is clean and not broken, the problem is likely in the hub, for example broken or stuck spring inside the hub. Fixing that is big professional repair (although there are some manuals and videos around and some questions on this site that could help with that).