Ugh. Stressful.
It’s not the first time, so don’t worry. This is a tiny bump in the road.
I agree – just ask the other teacher. Be open that you were having someone else do it, but her schedule is just too hectic and she suggested maybe finding another person.
A few tips for your new recommender:
Make it very easy for her to do the recommendation quickly. Have a quick one-pager to explain why you are going to boarding school, highlighting things you are talking about, maybe a list of ECs, or whatever. (Some people would call this a “brag sheet” but I would personally stay away from anything that sounds brag like, but more like “hey let me tell you about me because I don’t expect for you to have all this memorized since you are very busy.” I might even include a couple of examples from that teachers class, just to help jog their memory if they need an example or two.)
Note the recommendation forms are actually quite short (sadly short, IMO). so it really isn’t a crazy thing for someone to turn around quickly. So yes a very nice gesture to be grateful for, but you aren’t asking for the world. It should be doable in a matter of days I would think.
Lastly, I do not think you have to ask the other teacher to cancel the request. I think you just re-send a request to a new person. But not 100% sure.