Since you have an old car battery, the cost of damaging it by overcharging isn't much ($40?) and that would only happen if it got a lot of sun over weeks. A small home solar system might have thousands of dollars of batteries, so standard practice has long been for those to always have charge controllers.
For small systems the cost of a charge controller used to be significant so it was common to forego it in favor of simplicity. However, there are now very - inexpensive - controllers which no longer as easily justify the old practice.
If you can afford an additional $25, you might as well do it right. Also, it would come with instructions on what to do.
I was headed this way until about 2009. In the most modern and expensive setups you might have all the devices communicating and coordinating. In most, each one has its charging parameters that it's trying to meet, and the only communication is the voltage on the bank.
With multiple, uncoordinated charge controllers you may exceed the optimum charging current, and you should expect to have disagreements among uncoordinated charge controllers (and a water turbine with no charge controller) as their charging set-points and algorithms are almost certainly different.
If you have the money to put into it, or you see it as saving money on the battery bank's lifetime, a set of networked, coordinated smart charge controllers would certainly work more optimally (and can all work at the same time) - but you should also realize that any system that won't leave you in the dark for large parts of the year is going to have some excess capacity when everything is going well, charging-wise.
You can't simply add more battery without making it really hard to get it fully charged and equalized, and you can't reduce power sources without finding yourself in the dark after a few dry, cloudy, windless days. So sometimes, you are going to be dumping excess power. If you can do something useful with that power, great; (laundry is a good one, often) otherwise it's just part of the cost of having power more of the time. In the case of a fuel burning generator and a wind turbine, common sense would be to shut down the fuel-burner on a windy day.
Best Answer
Well, it's either dead, and you need a new battery, or it will recharge and perhaps you don't need a new battery. Since the typical alarm is very "trickle" rather than active charge, it may take 2-3 days.
Depending on what its built in-protections for the battery are, a complete discharge might have damaged it (if no low voltage cutoff - lead-acid batteries HATE being anything like "fully discharged" - they are not NiCads) or not.
Alarm batteries are typically designed to protect in case of a power outage for a certain period of time - if the time yours lasted was particularly short, it probably needs to be replaced. You could check the actual voltage with a meter to see if it's alarming (11.99 or less; or 12.5 or less, depending who you ask. 10.5 and under are pretty clear indications that one or more cells have definitely given up the ghost) and if it rises before you check it again.
Unclear why you chose to do this - there are not very many reasons to, and several not to. Despite being nominally rechargeable, there's a rather small finite life to these batteries, and deeply discharging them eats into that.
Common recommendation is to plan on replacing on a 3-5 year cycle (how well the alarm "checks" the battery varies between alarms, and it may or may not give you any indication that the battery is useless before you have a 5 minute outage and the alarm goes belly up in 3 or 30 seconds - which can be one reason to unplug for a couple of hours once in a while, but not to the point of killing the battery unless it can't make it for a couple of hours and you replace it.)