I was (have been being) tied up by (with my) work and studied (study) all these years so I didn't have enough time to cook, so (and finally) I decided to join (take part in) an Italian cuisine workshop. It seems to me that "tied up by work" should work, but I don't think we say it very much. We are more likely to say "I'm tied up; I have a lot of work". Also maybe it's better to use "enough time to learn how to cook" because the sentence suggests that the narrator wants to learn how to cook.

Hi Narges! You'll want to remove the extra word "being". So: "I have been tied up with my work and study all these years"...

I was (have been being) tied up by (with my) work and studied (study) all these years so I didn't have enough time to cook, so (and finally) I decided to join (take part in) an Italian cuisine workshop. It seems to me that "tied up by work" should work, but I don't think we say it very much. We are more likely to say "I'm tied up; I have a lot of work". Also maybe it's better to use "enough time to learn how to cook" because the sentence suggests that the narrator wants to learn how to cook.