Sounds like your battery is toast from being drained dead and left that way. Letting a car sit with the battery cables hooked up for a month can do that. Charge and load test or take the battery to a parts house and they will most likely do that for free.

Once you have a known good battery in the car, put a meter to the electrical system and check voltage while running. You should get around 14 to 14.5 volts with accessories off and the engine a bit above idle. A little lower at idle is acceptable. Alternators don't usually fluctuate between good and bad, but regulators sometimes do.

Heat. On a W124 I'm assuming this has ATC. Good luck.

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Whoever said there's nothing more expensive than a cheap Mercedes never had a cheap Jaguar.

83 300D Turbo with manual conversion, early W126 vented front rotors and H4 headlights 401,xxx miles
08 Suzuki GSX-R600 M4 Slip-on 26,xxx miles
88 Jaguar XJS V12 94,xxx miles. Work in progress.
99 Mazda Miata 183,xxx miles.