The archer wins

The caster gives up concentration on their first spell the moment they start to focus on a new concentration spell, thus the first ends before the second is complete.

The sequence would be:

1. The caster is concentrating on Greater Invisibility
2. The caster drops concentration on Greater Invisibility to cast Create Bonfire
3. Since the caster is no longer concentrating on their first spell, they become visible
4. The archer see can now see the caster and fires
   1. Assuming the readied Action was, "As soon as I see the caster, I launch an arrow"
5. If the archer misses, then the caster can finish their spell uninterupted

The archer wins

The caster gives up concentration on their first spell the moment they start to focus on a new concentration spell, thus the first ends before the second is complete.

The sequence would be:

1. The caster is concentrating on Greater Invisibility
2. The caster drops concentration on Greater Invisibility to cast Create Bonfire
3. Since the caster is no longer concentrating on their first spell, they become visible
4. The archer see can now see the caster and fires
   1. Assuming the readied Action was, "As soon as I see the caster, I launch an arrow"
5. If the archer misses, then the caster can finish their spell uninterrupted

See also this Sage Advice where a readied spell can interrupt another spell caster.