Does a surgical extraction for a partially broken wisdom tooth take much longer than a simple extraction and can it be done under local anesthetic only? are the type and amount of local anesthetic used the same as for a simple extraction?

Possibly. No two extractions are exactly alike. The procedure can be done under local anesthetic. Regarding how much anesthetic and the type, it is usually the same, but may vary slightly based on your specific circumstances. Either way, the tooth is coming out and you're not going to feel any pain.

Length of time depends degree of difficulty. Surgical extraction takes more time than simple extraction. Local anesthetic choice made by the Oral Surgeon depending on parameters of the procedure and patient's health history. This is a discussion to have with your Surgeon after an individualized examination.