The description of the Ready action (p. 193) states that you have until the start of your next turn to use a readied action. It continues:
When you ready a spell, you cast it as normal but hold its energy, which you release with your reaction when the trigger occurs. To be readied, a spell must have a casting time of 1 action, and holding onto the spell’s magic requires concentration (explained in chapter 10). If your concentration is broken, the spell dissipates without taking effect.
The spell is treated as being cast at that point when you declare you are readying the spell with whatever trigger is appropriate. Therefore, any restrictions that apply to other actions are based on the spell you ready, a cantrip or a spell using a spell slot. You are now also concentrating on the spell.
If a cantrip is readied, a bonus action spell can be cast as long as it does not require concentration (as you are already concentrating on the readied spell).
If a spell cast using a slot (i.e. a non-cantrip) is readied, you cannot cast a bonus action spell as per the bonus-action spellcasting rules (p. 202):
You can’t cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
You would not be prevented from casting a reaction spell. However, doing so would prevent you from being able to release the readied spell - as doing so requires you to have a reaction available, and as per the Ready action, it must be used before the beginning of your next turn.