First: a Spinner can be a very effective weapon (just look at current battle bots champion Tombstone or a lot of others in smaller weight classes as well), but (because of that) also very dangerous. No idea what kind of arena you guys will use, but without some really good walls around it (and a roof) i'd just avoid spinner for safety issues.
In this weight class the energy that a spinner should store to be effective is higher than the energy of a bullet (9mm) from a typical handgun. And that's just the minimum, it could get a lot more. you just don't want shrapnell's like that into you or any other bystanders.

But aside of that, i'd guess your drones etc. are brushless?
The usuall 12v geared motor is not, and a brushed motor won't work with a brushless ESC (unless you do some tinkering with the software etc.). Also, as far as i am aware most brushless esc from flying applications are made to make the motor go just one direction. But for a bot you'd want it to go both ways.

An additional gearbox won't be needed in most cases, but depends on the gearing on those, the rpm of the motors, the size of the wheels and how fast you want to be.
https://wahiduddin.net/calc/calc_speed_rpm.htm
This could help you, even if it is with imperial units. I've heard 10mph would be okay, but for a wedge going faster wouldn't hurt (if you can control it, but fpv racing should build up reflexes ). Just make sure your motors are strong enough, after all you'll want to push and shove your enemy. won't happen if those are too weak.
If you can't find two strong motors, maybe four weaker ones are the way to go. But generally for a wedge i'd make two wheels, and those stick out to the back, and also to the top- and bottom side of the bot, but not to the sides. Gives you some protection, but also the ability to drive away when someone gets under you.


But aside from that, 3 weeks for a wedge is absolutely possible, if you don't get problems with parts not arriving or stuff like that.

Make a scetch, arrange your internals in that (Battery, Motors, Gearboxes, ESC(s), Receiver, (if needed by your rules a removable link, switch or whatever to turn it on/off) and the wheels.
measure and/or calculate how heavy that stuff is.
Draw the shape of the whole bot, and calculate the surface area of those walls, top and bottom plate (in cm²).
Now multiplicate that with 0,3 (thickness) * 7,9 (density), and you know how much steel armour would weight at least. Or multiplicate with 2 (thickness) * 1 (density) to see if HDPE would fit better. Or 1*2,71 for Aluminium. Or...
Or like that with any other material, depending on the weapon types your enemy might use you'd maybe want to use metal for top and bottom (to make the wedge thinner and keep the edge sharp) and anything for the sides.
Also, these thicknesses are rough estimations. you can likely go with a lot more on a wedge, and depending on the general type of enemy you might want to vary this.

Oh, and try to put your ESC etc. on a layer of lexan or stuff like that to absorb a few shocks. Electronics don't like to get punched around.

Now someone with a bit more experience than me may correct me on this