There is an interesting 80 page pdf available online entitled :
UK Transgender law . Factsheet 01. Changing names and legal gender, put out by Press for change.
It gives all the ins and outs of employment, marriage and Criminal Records.
I’m happy to report that people on the sex offenders register can’t change the Sex on their criminal records .
But, they can request that police held records of non sexual offences be changed to their new Gender/Sex.