I would first start by looking at your map and picking a suitable alternate. Since hopefully you know your position already on that map, you can make a generalized turn towards the airport. Now, along the way you can start the calculations...

I guess if he just wants you to use your head and no other fancy instruments, try this:

Take a piece of paper and put it along an imaginary course line from your present position to your diversion airport. Then place a hatch mark at your present position and one at your diversion airport. This will get your distance. Then bring that paper over to any VOR rose printed on the map. This will get your magnetic heading, uncorrected for winds. Knowing the winds, this will help you correct the head you're flying for more precision. You can now take that paper and bring it down to the legend where you will find the NM scale and use it to find the approximate distance. Lets say its 50NM. Also, lets say that your flight-plan calculated ground speed was 130 knots, and now this turn takes you away from some of your tailwind component (or makes your headwind component increase...). You think you'll loose about 5-7kts of ground speed, but lets be conservative and say 10 knots. Now you estimate your groundspeed at 120 knots.

You know in your head that 120kts is about 2 miles per minute ((120 kts/hr)/60 mins)=2 NM/min, so you'll be traversing that 50 NM in about 25 minutes to the alternate.

Just shy of half an hour right? So take your GPH flow, and just shy of half would be 5.6 gallons.

You've found time, distance, and fuel consumption to your alternate.

Its no FMS solution, but it will get the job done.