<button class="button" @click="countDown">
 {{content}}
</button>

...

data () {
  return {
   content: '发送验证码',  // 按钮里显示的内容
   totalTime: 60      //记录具体倒计时时间
  }
},
methods: {
  countDown() {
    let clock = window.setInterval(() => {
      this.total--
      this.content = this.total + 's后重新发送'
    },1000)
  }
}

countDown () {
 this.content = this.totalTime + 's后重新发送' //这里解决60秒不见了的问题
 let clock = window.setInterval(() => {
  this.totalTime--
  this.content = this.totalTime + 's后重新发送'
  if (this.totalTime < 0) {     //当倒计时小于0时清除定时器
    window.clearInterval(clock)
    this.content = '重新发送验证码'
    this.totalTime = 60
    }
 },1000)
},

data () {
  return {
   content: '发送验证码',
   totalTime: 10,
   canClick: true //添加canClick
  }
}

...

countDown () {
 if (!this.canClick) return  //改动的是这两行代码
 this.canClick = false
 this.content = this.totalTime + 's后重新发送'
 let clock = window.setInterval(() => {
  this.totalTime--
  this.content = this.totalTime + 's后重新发送'
  if (this.totalTime < 0) {
   window.clearInterval(clock)
   this.content = '重新发送验证码'
   this.totalTime = 10
   this.canClick = true  //这里重新开启
  }
 },1000)
}

<button class="button" :class="{disabled: !this.canClick}" @click="countDown">
...
.disabled{
 background-color: #ddd;
 border-color: #ddd;
 color:#57a3f3;
 cursor: not-allowed; // 鼠标变化
}