I think I finally nailed it :-)

I have had difficulties with the concept of a relay ring oscillator because relays don't behave like transistors. With this kind of inverting gates, you just use odd numbers of gates with capacitors and that's it.

For relays it must be different. The trick is to use only one inverting gate (at the right) which is normally closed (N.C.). The others are non-inverting gates (N.O.) which add more delay.

It's going to be pretty clickety and remove concerns about bounces on the contacts. It can also control frequency and duty cycles easily, by adding more delay gates, making a more efficient overall design. For example it can generate a clock signal with 1/4 duty cycle, so the series lamp gets turned on during less time, consuming less and allowing faster operation.

Update (20160509): it works !

It is difficult to record the sound, I hope you can hear it.

With 3×(3300µF/22Ω), the clock period is around 600ms, or 1.6Hz, which is reasonable (not too fast, not too slow).