It may be a bit cheesy, but the familiar does not have to ready an action, you just state that the familiar is aiding
your next attack. Doesn't matter if 3 other PCs attack in between.
Under Combat Actions, Help
Alternatively, you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally's attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage.
I don't hit this in my game because it's just easier to have familiars and sidekicks go on the same initiative as the PC. I just have people choose which is going first and they need to stick with it unless they want to change before combat starts for some reason. I prefer to go with simplicity and fun over strict reading of the rules.