Needs ts-loader and sass-loader.

DateTimePicker.vue

<template>
<div class="datetimepicker">
    <div class="view">
        <input type="text" class="preview" :value="previewText" readonly>
        <button class="change-button" @click="openPicker">⏰</button>
    </div>
    <transition name="fade">
        <div class="picker" v-if="shown">
            <div class="date">
                <div class="month-selector">
                    <button @click="moveMonth(-1)">&lt;&lt;</button>
                    <span class="current">{{ year }}年 {{ month + 1 }}月</span>
                    <button @click="moveMonth(1)">&gt;&gt;</button>
                </div>
                <table class="calendar">
                    <thead>
                        <tr>
                            <th class="sunday">日</th>
                            <th>月</th>
                            <th>火</th>
                            <th>水</th>
                            <th>木</th>
                            <th>金</th>
                            <th class="saturday">土</th>
                        </tr>
                    </thead>
                    <tbody>
                        <tr v-for="week in weeks" :key="week.reduce((a, c) => a + c)">
                            <td v-for="d in week" :key="d"
                                @click.stop="pickDay(d)"
                                :class="{ today: d === day }">
                                {{ d }}
                            </td>
                        </tr>
                    </tbody>
                </table>
            </div>
            <div class="time">
                <table>
                    <tbody>
                        <tr class="buttons">
                            <td><button @click="moveHours(10)">+10</button></td>
                            <td></td>
                            <td><button @click="moveMinutes(10)">+10</button></td>
                            <td></td>
                            <td><button @click="moveSeconds(10)">+10</button></td>
                        </tr>
                        <tr class="buttons">
                            <td><button @click="moveHours(1)">+1</button></td>
                            <td></td>
                            <td><button @click="moveMinutes(1)">+1</button></td>
                            <td></td>
                            <td><button @click="moveSeconds(1)">+1</button></td>
                        </tr>
                        <tr class="preview">
                            <td>{{ hours.toString().padStart(2, '0') }}</td>
                            <td>:</td>
                            <td>{{ minutes.toString().padStart(2, '0') }}</td>
                            <td>:</td>
                            <td>{{ seconds.toString().padStart(2, '0') }}</td>
                        </tr>
                        <tr class="buttons">
                            <td><button @click="moveHours(-1)">-1</button></td>
                            <td></td>
                            <td><button @click="moveMinutes(-1)">-1</button></td>
                            <td></td>
                            <td><button @click="moveSeconds(-1)">-1</button></td>
                        </tr>
                        <tr class="buttons">
                            <td><button @click="moveHours(-10)">-10</button></td>
                            <td></td>
                            <td><button @click="moveMinutes(-10)">-10</button></td>
                            <td></td>
                            <td><button @click="moveSeconds(-10)">-10</button></td>
                        </tr>
                    </tbody>
                </table>
            </div>
        </div>
    </transition>
</div>
</template>

<script lang="ts">
import Vue from 'vue';

type DateTimePickerData = {
    picked: Date;
    shown: boolean;
    weeks: number[][];
};

export default Vue.extend({
    data(): DateTimePickerData {
        return {
            picked: new Date(),
            shown: false,
            weeks: [],
        };
    },

    computed: {
        previewText(): string {
            return this.picked.toLocaleString();
        },
        year(): number {
            return this.picked.getFullYear();
        },
        month(): number {
            return this.picked.getMonth();
        },
        day(): number {
            return this.picked.getDate();
        },
        hours(): number {
            return this.picked.getHours();
        },
        minutes(): number {
            return this.picked.getMinutes();
        },
        seconds(): number {
            return this.picked.getSeconds();
        },
    },

    watch: {
        picked(newDate: Date, oldDate: Date) {
            this.$emit('changed', newDate);
        }
    },

    methods: {
        openPicker() {
            this.buildMonth();
            this.shown = !this.shown;
        },
        buildMonth() {
            this.weeks = [];
            const allDays = new Date(this.year, this.month + 1, 0).getDate();
            let dow = new Date(this.year, this.month, 1).getDay();
            let target: number[] = [];
            for (let i = 0; i < allDays; i++) {
                target[dow] = i + 1;
                if (++dow === 7) {
                    dow = 0;
                    this.weeks.push(target);
                    target = [];
                }
            }
            if (target.length !== 0) {
                const fill = 7 - target.length;
                for (let i = 0; i < fill; i++) target.push(undefined);
                this.weeks.push(target);
            }
        },
        pickDay(d: number) {
            if (!d) return;
            this.picked = new Date(this.year, this.month, d, this.hours, this.minutes, this.seconds);
        },
        moveMonth(n: number) {
            this.picked = new Date(this.year, this.month + n, 1, this.hours, this.minutes, this.seconds);
            this.buildMonth();
        },
        moveHours(n: number) {
            const oldMonth = this.month;
            this.picked = new Date(this.year, this.month, this.day, this.hours + n, this.minutes, this.seconds);
            if (oldMonth != this.month) this.buildMonth();
        },
        moveMinutes(n: number) {
            const oldMonth = this.month;
            this.picked = new Date(this.year, this.month, this.day, this.hours, this.minutes + n, this.seconds);
            if (oldMonth != this.month) this.buildMonth();
        },
        moveSeconds(n: number) {
            const oldMonth = this.month;
            this.picked = new Date(this.year, this.month, this.day, this.hours, this.minutes, this.seconds + n);
            if (oldMonth != this.month) this.buildMonth();
        },
    },
});
</script>

<style lang="scss" scoped>
.datetimepicker {
    position: relative;
    margin: 8px 0px;
    .view {
        display: flex;
        .preview {
            font-size: 24px;
            margin: 0px;
            padding: 0px;
            height: 28px;
        }
        .change-button {
            font-size: 24px;
            height: 32px;
            margin: 0px;
        }
    }
    .picker {
        position: absolute;
        display: flex;
        justify-content: space-between;
        flex-direction: row;
        align-items: center;
        top: 36px;
        z-index: 100;
        background-color: #eee;
        border: solid 1px #666;
        border-radius: 4px;
        .date {
            margin: 8px;
            width: 240px;
            .month-selector {
                display: flex;
                flex-direction: row;
                .current {
                    flex-grow: 1;
                    text-align: center;
                }
            }
            .calendar {
                width: 100%;
                border: solid 1px #aaa;
                border-collapse: collapse;
                text-align: center;
                background-color: #fff;
                td,
                th {
                    border: solid 1px #aaa;
                    empty-cells: show;
                    cursor: pointer;
                }
                .sunday {
                    color: #f00;
                }
                .saturday {
                    color: #00f;
                }
                .today {
                    background-color: #faa;
                }
            }
        }
        .time {
            text-align: center;
            margin: 8px;
            margin-left: 0px;
            .buttons > td {
                margin: 0;
                padding: 0;

                button {
                    width: 100%;
                    font-size: 12pt;
                }
            }
            .preview {
                font-size: 24pt;
                line-height: 1.2em;
            }
        }
    }
}

.fade-enter-active, .fade-leave-active {
    transition: opacity 0.2s;
}
.fade-enter, .fade-leave-to {
    opacity: 0;
}
</style>