You could also go the opposite direction. The enemy ship refuses to move, citing a reason why they shouldn't let the good ship pass (spying/sabotage/weapons transport, take your pick). This creates a stand-off with parallels to the Cuban Missile Crisis, but as a background to the present race-against-time to save the crew of the crashed ship.
It also makes narrative sense, as cold wars tend to have large amounts of saber-rattling, with the occasional fate-of-the-world standoffs in between. This would achieve the delay you need, while potentially setting up other plot lines
Depending how long the stand-off lasts(Cuban Missile Crisis lasted 13 days), the location of a crash site(planet, moon with atmosphere), and damage inflicted by the crash, a number of things can happen:
Scenario 1 (2-4 week standoff)
- The crashed ship could have lost comms, but retain life support and supplies, with lots of duct tape emergency repairs by surviving crew
- The enemy ship and the good ship saber-rattle, and fire warning shots as they get closer.
- They each call for resupply, and already have military ships backing them up because they would've been scrambling at the beginning of this standoff
Outcome 1: Close call, but ultimately ends well due to diplomacy
Scenario 2 (1-week standoff)
- Crashed ship suffers significant damage, and comms either crap out after the send a full status report and casualty list, or stay up to broadcast the struggle to their gov't/the greater space community. They could have someone important to the government whose dying to help force the call to run the blockade
- The enemy ship and the good ship stand off, the good guys getting ever closer to the red line drawn by the enemy ship.
- The cavalry arrives for both sides, and the good guys run through it to save the crew, and a small skirmish ensues
Outcome 2: Cold war briefly heats up, may start in earnest depending on the political climate and reaction of the community to the actions of each side