I made an auto-play progress for the slider using CSS. When the activity class is added to the pagination, I run a CSS animation that lasts as long as the slider autoplay delay. Everything will work fine if you don't switch browser tabs or minimize the browser. As soon as we switch tabs, the slider autoplay pauses and causes the animation to stop. Maybe someone knows how to affect it?

import Swiper, { Navigation, Pagination, Autoplay } from 'swiper';

const heroSwiper = new Swiper('.hero__swiper', {
  modules: [Navigation, Pagination, Autoplay],
  autoplay: {
    delay: 5000,
    waitForTransition: false,
    disableOnInteraction: false,
  },
  slidesPerView: 1,
  speed: 800,
  grabCursor: true,
  navigation: {
    prevEl: '.hero__navigation-button--prev',
    nextEl: '.hero__navigation-button--next',
  },
  pagination: {
    clickable: true,
    el: '.hero__swiper-pagination',
    renderBullet: (i, className) => {
      return `<button class="${className}">${(`0${i + 1}`).slice(-2)}</button>`;
    },
    type: 'bullets',
  },
});

&__swiper-pagination {
    position: absolute !important;
    top: auto !important;
    bottom: 12px !important;
    left: 50% !important;
    display: inline-flex !important;
    width: auto !important;
    transform: translateX(-50%) !important;
    pointer-events: none !important;

    .swiper-pagination-bullet {
      position: relative;
      display: inline-flex;
      width: auto !important;
      height: auto !important;
      margin: 0 24px 0 0 !important;
      color: #605647;
      font-size: 16px;
      line-height: 20px;
      background: none !important;
      border-radius: 0 !important;
      opacity: 1 !important;
      transition: 0.8s !important;
      pointer-events: all;

      &::before {
        position: absolute;
        top: 50%;
        left: 35px;
        width: 75px;
        height: 1px;
        background: rgba(#fff, 0.3);
        transform: translateY(-50%);
        visibility: hidden;
        opacity: 0;
        transition: 0.8s;
        content: "";
      }

      &::after {
        position: absolute;
        top: 50%;
        left: 35px;
        width: 0;
        height: 1px;
        background: rgba(#fff, 1);
        transform: translateY(-50%);
        visibility: hidden;
        opacity: 0;
        transition: 0.8s;
        content: "";
      }

      &.swiper-pagination-bullet-active {
        margin-right: 110px !important;
        color: #fff;

        &:last-child {
          margin-right: 0 !important;
        }

        &::before {
          visibility: visible;
          opacity: 1;
        }

        &::after {
          visibility: visible;
          opacity: 1;
          animation: pagination-progress 5s linear;
        }
      }

      &:last-child {
        margin: 0 !important;
      }
    }
  }


@keyframes pagination-progress {
  0% {
    width: 0;
  }

  100% {
    width: 75px;
  }
}

UPD Solution to the problem:

document.addEventListener('visibilitychange', () => {
  if (document.visibilityState === 'visible') {
    /* Since swiper is still running in the background and only restarts 
    the animation when the user returns to the tab, it was decided to delete the class. */
    if (document.querySelector('.swiper-pagination-bullet-active')) {
      document.querySelector('.swiper-pagination-bullet-active').classList.remove('swiper-pagination-bullet-active');
    }

    // Without timeout the css animation does not start
    setTimeout(() => {
      document.querySelectorAll('.swiper-pagination-bullet')[heroSwiper.activeIndex].classList.add('swiper-pagination-bullet-active');
    }, 10);
  } else {
    document.querySelector('.swiper-pagination-bullet-active').classList.remove('swiper-pagination-bullet-active');
  }
});

Thank you Xena Lesenkova