How to pass the value from parent component to child component on load in LWC?

Here parent component is passing account-name prop to child. You don't need connectedCallback() in child for the custom logic, rather you need renderedCallback(). More detailed documentation here

child.html

<template>
    <p>Helllo this is child</p>
    <p>{accountName}</p>    
</template>

child.js

import { LightningElement, api } from 'lwc';

export default class ChildComponent extends LightningElement {
    @api accountName;
    renderedCallback() {
        console.log('Hello this is in render');
        console.log(this.accountName);
        **add you conditional business logic here**
    }
}

parent.html

<template>
<lightning-card title="My Account Record" icon-name="standard:account">
    <template if:true={account.data}>
        <c-child-component account-name={name}>
        </c-child-component>
    </template>
    <template if:true={account.error}>
        {account.error}
    </template>
</lightning-card>

parent.js

import { LightningElement, wire, api } from 'lwc';
import {getRecord} from 'lightning/uiRecordApi';
import NAME_FIELD from '@salesforce/schema/Account.Name';

export default class TestSO extends LightningElement {
    @api recordId;
    @wire(getRecord, { recordId: '$recordId', fields: [NAME_FIELD] })
    account;

    get name() {
        return this.account.data.fields.Name.value;
    }
}

Make the following changes in parent component and it will start working.

Parent Component: JS

import STATUS_FIELD from @salesforce/schema/customObject__c.status__c;
import { getRecord } from 'lightning/uiRecordApi'

const FIELDS = [STATUS_FIELD];

@api recId
@track status;
@track error;
@track flagIndicatingDataHasBeenLoadedInVariables = false;

@wire(getRecord, {recordId: '$recId', fields: FIELDS}) details({error,data}) {
if(data) {
this.status = data.fields.status__c.value;
this.flagIndicatingDataHasBeenLoadedInVariables = true;
} else {
this.error = error;
}
}

HTML (calling child component)

<template if:true={flagIndicatingDataHasBeenLoadedInVariables}>
     <child-component status-value={status} >
     </child-component>
</template>

Explanation: We are setting a flag after the data is loaded. Only after data is loaded in the parent, it will be passed to the child component.