My seafront balcony was asphalted about 15 years ago (the builder is now dead). It was left for several weeks before the table & chairs were moved onto it, but has never been hard enough to avoid the furniture leaving depressions in it. Can it be covered with anything firmer or do I need to rip the whole thing up & start again?

You may check with the local contractor and see if another layer of asphalt could be laid.
At 15 years, I would assume it is getting in fairly bad shape .anyway.

But a local contractor will have to just inspect it, and give you estimates based on actual condition.

Small diameter table legs etc will always leave an impression in hot mix asphalt (similar to what furniture will do to even a hardwood floor), especially in a hot climate. Using plywood or other material under the furniture legs to spread out the load would have prevented the problem, but if you don't want to have to do that, replace it with concrete.
As regards leaving the surface of asphalt for 15 days before putting furniture on it, that makes no difference. Once hot mix asphalt is compacted and cools, there is no further hardening taking place. Full traffic loads often must be placed on asphalt roadways within an hour or less of it being placed, without damage if it has cooled. But high temperatures can always soften it. Concrete is a much better product for your application.