After giving the engine a wash the battery

Sounds more like a wiring problem with the harness going to the cluster or the circuit board in the cluster. Most components get power from the same source but have different ground circuits.
The bat light stays on if the alternator is not charging the battery, the oil light says on if there is no oil pressure, and so on.

The battery light would mean the alternator is not charging the battery, the engine light is either a check engine light for the computer or low oil pressure (red) and the tail lamp warning is for a bulb not working.

Yes if the engine cranks the starter and battery are out of the picture.
Someone will need to check for spark at the plugs and fuel pressure to find out which direction to take.

howdie sounds like their could be ground problem but put the alt, under load and see what it puts out,if the battery was changed look for clamps are not loose check your ground to motor wire for corrosion

Sound like a problem with the oil sending unit or ground wires connections, that drain your charge overnight. Check it to solved.

usually cars stall when running and the batterie is removed, indicating a bad alternator.
perhaps your batterie is bad

what is the voltage of the battery when running, voltage when turned off, voltage after headlights are on for 10 minutes. is the battery good? how do you know. these take a sealed gell cell battery.

battery is probably low.when trying cranking for a period of time battery get weak thats why battery light on.charge battery.to see if it hold a charge.when battery fully charged it should be 12.5 volts.with engine running alternator out put should be 13.5 to 14.5.if not replace the alternator.or have it tested.when replacing valve covers make sure your pvc valve.in valve cover in place if not your car will stall out and idle rough.your oil leak on inside of your car from a leaking oil line to oil pressure gauge.thats why your oil light is on.

Well...your battery is dead. Check the connections of the cables on the battery posts to make sure they are tight and clean (no corrosion). While you're under the hood, look at the belt (serpentine) that runs your alternator. Is it there? If it fell off, your alternator can't charge your battery. You may have a bad alternator, a bad battery, or no belt....but use a battery charger to charge the battery...your alternator is only designed to "maintain" the charge, not bring it up to full power. Good luck, alternators for the MX5 are spendy, so I hope it's one of the other problems.

its sound like our alternator belt broken that belt drive the water pump broken alternator belt charge light on belt dont turn water pump overheating dont drive with it check that out first thank you pierre