Nothing bad will happen. Most wild animals never take a bath. Their skin is more or less fine. If you take people - many northern peoples never had a tradition of taking a bath - nothing special happened to them either. In a sense, the skin will be even better; the use of a large amount of detergents, all kinds of balms, conditioners, etc. etc. reduces the ability of the skin to cleanse itself, deprives it of the characteristic lubrication secreted by the sebaceous and sweat glands, and it is useful in many ways.
But another question will arise - if you do not carry out regular hygiene procedures, you will have a very unpleasant by today's concepts, aroma. This will be a big problem for you - social interactions will quickly be minimized. You've probably seen how quickly people leave the last subway car, if it contains a person without a fixed abode, who has not been sanitized for a long time. Therefore, it would be quite reasonable to carry out hygiene procedures with sufficient regularity. But without fanaticism. At least taking a shower three times a day is more harmful to the skin than not taking a bath for a lifetime.
It's okay. humanity has lived in this regime for 100 thousand years. :) Some scientists believe that taking a bath too often is harmful. Monkeys never wash.
On the one hand, the skin gets dirty. On the other hand, the superficial dead skin layer is constantly peeling off.