I've designed a simple circuit that might do what you want, if you're still interested.
It accepts two dry contact inputs and provides two dry contact outputs. It needs a supply voltage of 12VDC.
The two relays are SPST (Form A) relays with 12VDC coils.
The run timeouts are set by the two trimpots in conjunction with C1 and C2. If you're working close to either end of the trimpot, you can adjust C1 and C2.
Each part of the circuit ("UP" and "DOWN") is independent and there is no interlock between them; if the UP and DOWN inputs are activated together, the outputs will be activated together too. Here is a circuit description for the "UP" circuit.
While the contact opens, C1 charges quickly via R1 and D1, and C1 is held charged while the contact is open. Q1's emitter is at the positive supply rail so Q1 is not biased, so Q3 does not conduct, so K1 is not activated.
When the contact closes, Q1's emitter is pulled to ground, and its base receives bias from C1 via R3. Q1 is a Darlington transistor so it requires very little base current. It conducts and energises Q3, activating K1 and closing the output contact.
C1 now discharges through R7. When C1's voltage drops below about 1.4V it is no longer enough to keep Q1 forward-biased, so Q1, Q3 and the relay turn off. The output will not activate again until the input contact has gone open, then closes again.