Using a milwaukee 18v cordless tool battery to power my 5520?

Since you must use a Dell AC adapter to complete the detection loop and charge the battery, the only way you'll do this is to connect the battery to a DC to AC inverter - and use the inverter to power the adapter.  You'll need much more than a single 18 V battery to do this, however -- you'll lose a lot in the DC to AC inversion.

Translation:  not worth the effort.

@svejkovat  Even if you can wire up something that won't damage your system or otherwise pose some sort of hazard, Dell systems rely on being able to identify the wattage of the attached power source, which only works with Dell AC adapters.  When the system can't identify the wattage of the power source, it treats it as an Unknown source, and in that case the system plays it safe by reducing its power consumption as much as possible in order to minimize the risk of trying to draw more power than the Unknown source is rated to provide.  Those measures to reduce power consumption will include charging the battery slowly or not at all, as well as significantly throttling CPU and GPU performance.  So it won't be a great experience.

Also, you provided the voltage output and the Ah capacity of the batteries, but not the amperage output.  The Precision 5520 is designed for a 130W source, which is 19.5 V at 6.67 A.  Of course if the system can't identify its power source, it probably won't even try to draw nearly that much amperage, so this is probably just academic in this particular case, but the fact remains that you can't just look at the voltage output and energy capacity of a power source.  You have to look at energy delivery rate.  I've seen multiple questions here from people wondering why they can't use their large, 20,000 mAh USB power banks to charge their laptops, and the answer is because although those batteries have high capacity, they're not designed for high wattage output.  They're designed for smartphones and tablets, not laptops.  I have no idea what those Milwaukee batteries are rated to provide in terms of amperage.

You can use this inverter for M18 Milwaukee batteries https://www.milwaukeetool.com/Products/Batteries-and-Chargers/Other/2846-20