When you unplug the battery does the m1 go dead? If the m1 is still working then your AC input is obviously working and the m1 is able to use it.
Measure the voltage of the wires that go to the battery with the battery disconnected. If it measures more than 13v then it should be able to charge. 0 volts either means the Elk's charger is not functioning or it cycles and is currently off. If you do get good voltage, connect the negative terminal of the battery to the neg Elk wire. Use a multimeter with amp settings and touch it's red lead to the Elk's red and the multimeter black wire to the battery red. If there is current, then the battery is being charged presently. If there is virtually no current in the presence of 13 plus volts, then you have a fully charged battery (or a completely shot battery). I am sorry but I don't know the exact numbers. Possibly the battery manufacturer has that data available in graph form online. If not, you should be able to look it up for a generic lead acid battery of similar size. Here is a reference. http://batteryuniversity.com/learn/article/charging_the_lead_acid_battery
The Elk probably has a battery maintenance program but I don't know what it is. In general, you don't want to keep a battery continuously charging after it is at full charge since it will shorten battery life. Chargers will measure the current going to the battery and from that know how charged the battery is. More current flowing to the battery equals a less charged battery. As the battery gets closer to fully charged the current drops. It is possible that Elk cycles the battery by only charging it for short periods of time once it hits a fully charged state. Probably it just drops the voltage to "float" level.
A new battery should come mostly charged. Every time I have put a new battery on an alarm it never gave me a low battery warning.