The scientific term for this process is senescence.

This word encompasses the aging of plants in general. It describes not only the phase that fruits and vegetables enter after their ripening phases, but also the process by which leaves turn colors in the autumn, etc.

This is the wikipedia article on the process of senescence.

The adjective form of the word is senescent.

Showing signs of the farmer's neglect, the fields were filled with overgrown vines and vegetables in various states of senescence.

picked past their prime (also) peak

• When harvested past their prime, beets have a strong taste and a tough, pithy texture"
  (Google Books)

• Here are some ways that you can use your frozen, past-prime fruits and vegetables.

• Tomatoes that are past their prime can be pureed and frozen for up to 6 months.
  (Frugal Living Mom.com)

• The Pilgrims may have eaten turnips, but why should we? Let's face it; the turnip can be a tricky vegetable. If they're even a day or two past their prime, you'll wind up dining on a veggie that tastes more like a piece of wood than an appetizing holiday side.
  (How Stuff Works.com)

• When florets [broccoli] on the outside edge of the head are large and full, this is another helpful indicator that you’re good to go. But don’t wait too long. When they begin to turn from green to yellow, this is a sign that they’re beginning to flower, and past their peak.
  (Gardener's Path.com)

A single-word alternative could be

overmature

overmature: past the age or condition of maturity
a : beyond the stage of desirable or optimal development or productivity

• overmature wax beans

• The squash is very tender and should be handled with care to prevent skin cuts and bruising. Avoid purchasing any with … a dull, shriveled skin, which is a sign of overmaturity.
  (Merriam-Webster)

Run to seed could also work in this context -

Run to seed

To become old and decrepit. Plants that are allowed to set seed after flowering either become bitter to the taste (lettuce) or will not bloom as well the following year (daffodils, tulips). Henry Fielding used the term figuratively in an essay of 1740: “For Virtue itself by growing too exuberant and . . . by running to seed changes its very nature.”

(thefreedictionary.com)

For a number of vegetables, the word you're looking for is bolted. Bolting is

the production of a flowering stem (or stems) on agricultural and horticultural crops before the crop is harvested, in a natural attempt to produce seeds and reproduce. These flowering stems are usually vigorous extensions of existing leaf-bearing stems, and in order to produce them, a plant diverts resources away from producing the edible parts such as leaves or roots, resulting in a poor quality harvest. Plants that have produced flowering stems in this way are said to have bolted.

My grandfather was a farmer; he said they were left "too long on the vine."

I searched—vegetables "too long on the vine"—to see if farmers (and the like) still said this. Google returned 415,000 results; several examples from the first page follow.

Re: peas (Burpee.com)

Pods left too long on the vine get tough and stringy.

Re: tomatoes (Omaha.com)

Large heirlooms can crack if left too long on the vine.

Re: cucumbers (DenverPost.com)

If left too long on the vine, they turn yellow, get bitter and seedy.

Re: string beans (wol.jw.org)

Staying too long on the vine also makes string beans “stringy.”

Re: melons (ChicagoBotanic.org)

For the home grower who left the melon too long on the vine or lost it to the critters, here's a positive spin: worms love melons.

Papa also said "past maturity" (but some may say overmature, of course, as previously suggested).

Also, overgrown is appropriate (per University of Illinois Extension) within its proper context.

Check the garden frequently for ripe produce during harvest time. Vegetables continue to grow and before long they are overgrown.

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US, SE Region