Learn Arabic (Modern Standard): ليس عندي وقت.

While ليس means "is not", Arabic treats sentences about having something as nominal (noun-based) sentences. عندي literally means "with me" or "at my place". By putting ليس in front of it, you are literally saying "time is not with me", which is how you say "I do not have time".

Why is it ليس and not لست (I am not) since the English translation says "I do not have"?

This is a very common trap! When you negate "having" something, ليس does not match the person who owns it. It matches the thing being owned, because grammatically, that thing is the subject of the sentence. Since وقت (time) is a masculine word, we use the basic masculine form ليس.

What are the exact parts that make up the word عندي?

It is built from two pieces: the preposition عِند meaning "at" or "with", and the attached suffix ي meaning "me" or "my". Together they form "with me", which is the standard way to express "I have" in Arabic.

Why doesn't وقت have the word for "the" (ال) attached to it?

Just like in English, we leave off the word for "the" because we are talking about time in a general, unspecified way. If you said ليس عندي الوقت, it would mean "I do not have the time" (referring to a specific scheduled time or moment).