It is very rare to have a serious allergic reaction to a flu vaccine. If this does happen, it usually happens within a few minutes of having it done3. The person who vaccinates you will be trained to deal with allergic reactions. Some flu vaccines are made using eggs. This means that if you have an egg allergy, you may be at risk of an allergic reaction to the flu vaccine injection5. Ask your GP or pharmacist for a low-egg or egg-free vaccine.
3 Flu Vaccine - NHS (www.nhs.uk)
5 Green Book Chapter 19 Influenza (publishing.service.gov.uk)