There are two relevant rules in the player's manual, the focus being on relevance:
Some special abilities, spells and situations allow you to take a special action called reaction. […]
When you take a reaction, you can not take another one before the start of your next turn. If the reaction interrupts the turn of another creature, that creature may continue its turn just after the reaction.
-PHB, p. 190
If you are surprised, you can not move or do anything in your first round of combat, and you can not react until the end of this round. A member of a group may be surprised even if the other members are not.
-PHB, p. 189
Thus, between these two sections, we understand that the only point where a reaction is "refreshed" is at the beginning of a character's turn; there is no explicit point where the "initial reaction" is given, so it is available before the character has his turn.
The only scenario in which a character would not gain his reaction is if he is surprised or otherwise under an explicit effect that prevents the use of the reaction of a character.
A character needs an action that allows him to use his reaction
Most of the regular uses of Reactions are unaffected because they happen before the first turn of a character (Attacks of Opportunities, Counterspelletc.), but one of the common uses is prepared action; According to the rules, a character will not be able to react with a ready action because they could not put it in place first because of not having had their first round yet.
Strictly RAW, "Actions in Combat" can not be taken apart from the initiative. Many DMs drop this restriction on a case-by-case basis, as it would be tedious to rule on this rule for every possible decision that a character might make, but as a general rule, allowing characters to act out of combat may in good standing General. drive the characters to additional attacks, and is a concept that is meant to encapsulate by order of initiative.
So, unless otherwise stated by your DM on the Ready action, a character will not be able to use the reaction given by Ready Action before his first turn.