2005 Vauxhall Astra car reverse gear not working, moves forward in neutral

Yes why not try replacing the radiator and flushing the box and replacing the fluid first ,that should not cost that much and may fix it.

the auto box will be destroyed if water has got in replace rad the cooler for the autobox is in the bottom of it.
you can get the autobox rebuilt at various places eg gloster transmissions in bristol etc you will be looking at about £1000 to £1500 still expensive but better than 5k.take car to one of these places and they will quote.

Have you had a look at this similar problem: Car reverse gear not working, moves forward in neutral

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My car an Astra 1.8i Auto 2005 reg. with 53k miles on the clock began by loosing water just a little bit (0.5l) in a 6 month period. I could not trace the water loss. Last month suddnly the revedrse gear started jerking and refusing to engage. This was followed by the car getting in safe driving mode and refusing to do a change down. When I finally drained the ATF which was very dark and muddy. I fushed the system but did not change the radiator. 2 weeks later the car stalled refusing to engange any gears a all. Vauxhall (Vx) have quoted me a £4000 to replace the gearbox. I a absolutely horrified that Vx have such a car design flaw in a modern car and they have no intention of dealing with it to salvage the reputation of their brand. It's appalling. I have now read scores of reports where similar cars had their gearboxes fail at the same milage. Shouldn't we expert an auto box to last at least 100k miles with proper service?