DateTimePicker.vue

<template>
    <div class="date-time-picker">
        <datepicker v-model="date" :inline="true" />
        <vue-timepicker v-model="time" format="hh:mm A" :minute-interval="15" />
    </div>
</template>
<script>
import VueTimepicker from 'vue2-timepicker';
import Datepicker from 'vuejs-datepicker';

export default {
    name: 'date-time-picker',
    data: () => {
        return {
            date: new Date(),
            time: {
                hh: '08',
                mm: '00',
                A: 'AM'
            }
        }
    },
    components: {
        Datepicker,
        VueTimepicker
    },
    props: {
        value: Date
    },
    methods: {
        emitDate() {
            const timeString = `${this.time.hh}:${this.time.mm} ${this.time.A}`;
            const dateString = this.date.toDateString();
            this.$emit('input', new Date(`${dateString} ${timeString}`));
        }
    },
    watch: {
        date() { this.emitDate() },
        time() { this.emitDate() }
    }
}
</script>

SomeOtherComponent.vue

<template>
  <date-time-picker v-model="date" />
</template>
<script>
import DateTimePicker from './DateTimePicker.vue';

export default {
    name: 'some-other-component',
    data: () => {
        return {
            date: new Date()
        }
    },
    components: {
        DateTimePicker
    }
}
</script>