Ok, here's what you do:
Let's call the bigger enemy civ Civ A, the other Civ B.
Sign a mutual protection pact AND a Right of Passage pact with civ A, if you can.
Land your gigantic army of tanks (or Modern Armor) and as many nukes as you can get (use nuclear subs to ferry them) on Civ A's land.
Try to get them Civ A to declare war on Civ B. (try to get Civ B to declare on you first by infuriating him even more)
Wait one or two turns. Position your tank armies (plus a bit of artillery) near Civ A's cities and also position your nukes near some other cities (with 1 mech infantry and one or 2 tanks/modern armor standing by on high ground, a tile away from cities so as not to incur blast damage)
Also try to block Civ A's offensive army with some mech inf. conscripts (which should be in Civ B's territory)
Nuke Civ A's cities. (preferrably from the coast, one tile away) Take them (there will still be some resistance) Take other, weaker, cities as you can. Try to not overextend yourself.
Fortify, heal, and take other cities. Bring in more nukes if you have to.
If you play your cards perfectly and have about 2 tank/modern armor per enemy city (some tanks/modern armor, depending on difficulty level, should be in armies), plus nuke 60% of them, you should be able to take over Civ A in abut 4-6 turns (if, say, enemy civ has 18 cities)
the KEY here is don't overextend yourself.
Now kill the other Civ or make peace (which will prolly be impossible because you just reneged on three very important diplomatic agreements and nuked prolly 12 enemy cities

), and wait for more nukes.
