I am having double eyelid surgery under local anaesthesia & was wondering about the process when they insert the needle?

See below. Most times this procedure will also use a sedative so you will not be aware of the needles. Please talk to your surgeon to ask how they do it.
Local anesthesia . They will use a very small gauge needle and slowly inject the numbing agent. You can expect 30 seconds of burning afterward. Consider asking for some anti-anxiety medicine to take by mouth before surgery.
Sedation. If the anesthesia is strictly local, then the surgeon will give the local into the eyelids. If there is to be mild sedation also, then a surgical nurse or an anesthesia provider may give you a little sedation by way of an IV line prior to the surgeon giving you the injections. Talk with surgeon before the procedure to discuss what options are available and which will be best, and safest, for you.
Eyelid Anesthesia. I perform almost all of my eyelid surgeries under local anesthesia with either oral (pill) or intravenous sedation. Typically, i wait until the patient is very comfortable relaxed and even asleep. With a very small needle and gentle injection, the local anesthesia is usually only slightly uncomfortable but afterwards, surgery is very easy for the patient.
Not a big deal... Local anesthesia for eyelid surgery is almoat always accompanied by some type of conscious sedation, i.e. Medication given to relax you and make you mildly drowsy. The lids are anesthetized with a very tiny needle; most patients tolerate this very well.