The sprocket looks pretty well fucked in that picture. I'd wager that if you put a new chain on it, even reverse, it will sound like a can full of spanners.
>> Should I replace the chain or continue to use it? Obviously it doesn't skip but the worn chain will accelerate ring and sprocket wear.
It's up to you. You could change the chain and sprocket now and the chainring should be fine for many years more. Or you could "run it into the ground" and change the whole lot in a few years' time.
>> What happens if I continue to use the chain?
Not much. As you say it won't skip or unship. The chance of the chain breaking obviously gets greater the older it is, but that's a rare occurrence.