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This appears to be bacterial blotch. Most likely, the soil remained wet for too long or did not dry out thoroughly between waterings.

You can attempt to correct with a fungicide, but since they are annuals they will probably not recover. Most likely, you will need to replant in fresh soil. You will want to wash your container out very thoroughly, also.

Here are a couple of articles that will help:

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

https://www.gardeningknowhow.com/ornamental/flowers/chrysanthemum/growing-chrysanthemum-flowers.htm

Typically, you will see the new growth of mums in early spring. If you didn't accidentally pull up the mum roots when you planted the begonias, I would think the mums would come up around them.

Some mum varieties only come back a few years, others not at all. So don't be dismayed if yours didn't return this year.

Here is more information about the lifespan of mums:

https://www.gardeningknowhow.com/ornamental/flowers/chrysanthemum/chrysanthemum-lifespan.htm

Your mums look lovely! Since your window box is lattice lined with a coco liner, as opposed to a solid container, your plants are more susceptible to drying wind and high temperatures. You will need to check the soil once or twice daily and water if the the soil is dry.

Also you mentioned the pot size. If larger pots will fit in the window box that you could transplant into, that also would give your mums more protection against the elements.

And if you have a garden bed in full to part sun it would be fine to plant your mums there. If you have a deep edging around the bed, it should keep mums in and grass out.

Here is more information.

https://www.gardeningknowhow.com/plant-problems/environmental/heat-stress-plants.htm

You will have to ask your local providers. We do not sell plants. Thank you.

If mums return, they usually have new growth sprouting from the bottom of the plant. The top twigs are pruned off. If you don't see any growth by now, you can consider it lost.

The mums planted in fall are often treated as annuals. Spring planted mums, hardy mums, are more likely to return the next year.

These articles may help:
https://www.gardeningknowhow.com/ornamental/flowers/chrysanthemum/growing-chrysanthemum-flowers.htm

https://www.gardeningknowhow.com/ornamental/flowers/chrysanthemum/chrysanthemum-lifespan.htm

This is going to be a devastating answer, unfortunately. Verticillium Wilt is likely the culprit, and cannot be cured easily.

Here is an article that will help:

https://www.gardeningknowhow.com/plant-problems/disease/verticillium-wilt-treatment.htm