What's a simple word for "un-split" or "made of a single piece"?
Solution 1:
The table is undivided but the word has unnecessary overtones of divisions or compartments, which I do not think relevant to your specification. I find no term appropriate among the many synonyms of undivided or the antonyms of separated and divided; they relate too much to the concept of joining of parts.
Consequently I suggest extending the use of the easily understood term one-piece from its often limited application to clothing to your table.
one-piece = a piece of women's clothing that is worn when swimming or on a beach and consists of a single piece of material rather than a separate top and bottom
Interestingly, the same link in Cambridge dictionary gives a relevant example that extends the given definition in the way that you need:
"In contrast to all other devices presently being tested, this one-piece device may be easily retracted and repositioned until the time of eventual release"
Cambridge dictionary
Edited to account for the union of separate pieces:
It might unified or even unitary, both as defined in Cambridge dictionary
unitary example:
The details, the technological and formal solutions were drawn into a synthesis and their many elements into a unitary order.
Cambridge dictionary
Forming a single or uniform entity.
‘a sort of unitary wholeness’
Oxford lexico
Of these two "uni-options" I prefer unitary as describing the nature of it rather than what has been done to make it.