How dangerous are my worn rims? I took my bike for a service recently and was told it needed so many repairs that it was more cost effective for me to replace it.

One of the repair jobs was to replace the wheels due to worn rims. The shop told me they were so dangerous that they refused to replace the brake blocks - which were also dangerous - for fear of causing extra wear.

I am happy to replace the bike but this is a bad time to do so. I'm looking to move house and my eventual location may impact what kind of bike I buy. I'd like to ride on my worn rims until then.

The rims have a groove to indicate wear. The groove is still visible but it has become very shallow:



I've sorted the brakes for now by tightening the cable. But realistically how safe am I riding on these rims and for how long?
Rims are worn out when the groove disappears. If the groove is clearly visible, well-defined and of uniform depth all the way around, the rim is not quite worn out and certainly not dangerous.

New brake blocks will cause less wear as they will be free of hard particles of grit and metal that get embedded in the relatively soft rubber over time.

Given the repair shop's bad advice on the state of the rims and refusal to replace the blocks, I'd disregard all of their other advice about needed repairs. I suspect they just saw a well-used bike and decided they could persuade you to purchase a new one. Find a repair shop with a better reputation and get a second opinion.

Replacing and adjusting brake blocks yourself is relatively easy if you have basic tools. Park Tool has an excellent web page and collection of videos that walk you through the process.