Can I apply nodal analysis if the wire doesn't have a resistance?
Yes, you can. That is essentially how SPICE works.

If you're doing back of the envelope calculations, you'll likely be modelling D4 and D3 as voltage sources when they are in zener operation and forward bias operation respectively. In that case you cannot apply nodal analysis (nodal analysis doesn't apply to circuits with voltage sources) but you can apply modified nodal analysis.


  The way I learned is: \$\frac{V-V_{D4}-V_{D3}}{R}\$ but in this case there is no resistance across \$V\$ and ground. Should I put \$R=0\$?


What you should have learned is to write the equations for KCL at each node:

$$i_1 + i_2 + i_3 = 0$$

assuming \$i_1\$, \$i_2\$, and \$i_3\$ are the current of the three branches connected to node v, defined so they all flow in to the node. Then you need to figure out how to express these three currents in terms of the voltage at node v and the other nodes of the circuits. If the circuit elements were all resistors, you might get an equation like the one you used. But if the elements are diodes (neglecting zener or avalanche operation), you'd need to use the Shockley equation:

$$ I_D = I_s \exp\left(\frac{qV_D}{nRT}-1\right)$$

where \$I_D\$ is the current through a diode and \$V_D\$ is the voltage across the diode (something you'd be able to write as \$v_x - v_y\$ where \$x\$ and \$y\$ are two nodes in your circuit), and \$I_s\$ and \$n\$ are characteristics of the diode. To model the zener diode you'll need an even more detailed model that includes zener behavior.