Hi all, partner went out to her fiesta ( 63 reg 1.5 diesel) and she had a flat battery

We charged the battery for a day, put it back on the car and put a voltmeter across the terminals with the engine running, it read just under 12 volts, when not running its 12.2 volts
when running even with the lights and heater blower etc turned on the battery warning light doesn't come on

is there anything else we can check before buying a new alternator