"Just to check, if someone born with a penis gets a gender recognition certificate, have they had to have their penis cut off in order to do so?"
No - although 87% of trans women have wither had, or want vaginoplasty:
www.thetaskforce.org/static_html/downloads/reports/reports/ntds_full.pdf
... you don't need it to get a GRC. This is because some people either don't want surgery, or are unable to undergo surgery for medical or other reasons.
However, the complex and frankly rather marvellous micro-surgery involved is a little more than "cutting off the penis". In the hands of the best surgeons, like the top ones in Thailand, it's mainly a case of redistributing the soft tissues to return them to their original feminine configuration (we all start with essentially female genitalia).
So for example, the labia majora and the scrotum are formed of the same tissue. In the typical boy, the tissue fuses together along the raphe line to form a scrotum and on the typical girl it doesn't fuse and forms the labia majora instead. So, part of vaginoplasty, is to open the scrotum along the raphe line and give the tissues the typical female configuration (or rather "return them" to their original configuration) rather than the typical male one.
Similar thing with the penis/clitoris - they are the same organ that develops along different routes according to hormonal influence rather than different organs altogether like say a kidney and finger are.