SIM Cards should all break down to the nano size which is generally the same size as the chip on the SIM Card.
Are you sure you're inserting it into the correct side of the SIM tray?
Important to note also an iPhone does not support memory cards/SD Cards, so the tray is only for your EE SIM.
In order to transfer data, download the "Move to iOS" app on the Play Store on your old phone.
Find a guide here from Apple on transferring from Android: https://support.apple.com/en-gb/118670
