Regrouting is a labor intensive pain. Even if you use a multi-tool. There's no way I'd do it manually. And there's no way I'd waste the effort on that bath either.
What i'd do is hack, but i'd go get me some 6 mil poly and just tape it to the walls at the top and maybe add a bead of silicone to keep it watertight at the overlaps in the corner. Make the world's biggest shower curtain. It will keep behind the wall dry until the tearout occurs.
Then spend your time, money, and energy coming up to speed on how to do a proper bath reno with a proper waterproof shower. And move the reno date forward as much as the budget allows.
Do it right from the beginning if you only want to do the job once. It'll be cheaper in the long term.
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Although I do appreciate the "perfume on a pig solution", there's no way I would put the time and energy into regrouting this bathroom. I had to do this years ago on a small section of floor and it's no fun. I actually like the 6 mil poly "shower curtain" idea. Spend $30 or $40 on the box of poly and hang your "curtain". In the meantime, develop a remodel plan and arrange for the necessary financing to facilitate getting that bath fixed as soon as possible. If mold is a concern move sooner versus later.