The Fighter must wait until round 3
During round two in your scenario, when the Cleric heals the Fighter, the Fighter's turn has already ended for that round, so they would need to wait until the next round to take another turn.
The rules for death saving throws state the the saving throw is made at the start of the dying creature's turn (the Fighter, in this case):
Whenever you start your turn with 0 hit points, you must make a special saving throw, called a death saving throw, to determine whether you creep closer to death or hang onto life.
So this confirms that the Fighter must have started their turn. Then, the rules for dropping to 0 HP also state that:
When you drop to 0 hit points, you either die outright or fall unconscious
And an unconscious creature:
An unconscious creature is incapacitated, can't move or speak, and is unaware of its surroundings
And an incapacitated creature:
An incapacitated creature can't take actions or reactions.
Since the Fighter started their turn, made their death saving throw, was unconscious due to being at 0 HP, which means they are also incapacitated, which means they cannot take actions, that means they can do nothing on their turn besides making their death saving throw, upon which their turn ends. At that point the next in the initiative order is the Cleric, so as soon as the Cleric's turn begins, the Fighter's turn is ended and must wait until round 3 to take another turn.
NB: If it so happened that the goblin wanted to move away from the Fighter (and for some reason didn't want to take the disengage action), then the now-conscious Fighter would still be able to use their reaction to take an opportunity attack, since they are no longer unconscious and no longer incapacitated, and thus are able to use reactions again. This can be done during other creature's turns, but anything else would need to wait until round 3, as I explained above.
However, as @KorvinStarmast reminded me, such an opportunity attack would be made with disadvantage as the Fighter would still be prone, since he has not yet had an opportunity to stand up (which uses movement, which he would need to wait until his next turn to use).