The battery light is intended to tell you that the alternator is not charging the battery. If that is the case, the steering will go heavy, as it demands a high current.
A steering fault will not illuminate the battery light.
A common fault is the main battery earth cable, from battery, to body, then onto gearbox. These corrode internally, so can look fine, but struggle to pass current. A failure of this will cause the issue you describe.
It could of course be a simple matter of the alternator failing.
If after a normal start, the battery light goes off, but then comes on again, I'd be leaning towards the cable as the problem. Does the light illuminating coincide with any gearchange, moving away, or other action that might cause the engine to move on its mountings? That would definitely point to a cable breaking internally.
If you have access to jump leads, put one between the battery negative terminal, and the engine. If the problem goes away, the cable is at fault. A voltmeter across the battery will show if the alternator is charging. Should give around 14v with the engine running. If not, wriggle the earth cable, (under the battery, so be careful not to touch anything moving, or hot.) If the voltage changes, that again shows the cable as the problem. If the cable if fine, the alternator is the most likely problem.
The main power cable could be at fault, but this is less common. Check all connections, and inspect for cleanliness.
