Scrumplestiltskin here are the sources:
Trans prisoners = 60. This is a Government stat from the Ministry of Justice in response to Freedom of Information requests, and reported here: www.bbc.co.uk/news/uk-42221629
The same article says there are 125 trans prisoners but points out that this figure may be an underestimate, as it doesn't cover everyone in prison and may not include eg those on shorter sentences or who didn't declare they're trans. The article points that out as it's trying to refute Fair Play for Women's prior estimate of 41% of transwomen prisoners being sex offenders. But when we use the known number of sex offenders against population size, it means the 60 is a minimum - there may be more trans sex offenders who didn't declare themselves as trans / didn't do the MoJ survey.
TAs will throw an accusation out that the trans stats might not all be transwomen, they might be transmen. I read a justifcation for they're all transwomen somewhere but I can't remember where now. I think it was probably Fair Play for Women and based on the (male) prisons they're in.
Rates of male / female sex offenders in UK prisons.
These come from the download Prison population: 31 March 2018 on this page. www.gov.uk/government/statistics/offender-management-statistics-quarterly-october-to-december-2017
However I've just realised I probably shouldn't be using the 125 stat for women, as that's from the end of 2017 - but the only stats we have on trans prisoners (above) are from the beginning of 2017, so we should probably be using a figure from the same same period so it's a fair comparison (not that it makes any difference to the overall picture).
2017 Q1:
Male sex offenders: 13,132
Female sex offenders: 114
For completeness, here are the most up to date figures:
2018 Q1:
Male sex offenders: 13,431
Female sex offenders: 131
Both male and female sex offences are rising. However we don't know how many of the female figures are actually transwomen. It's perfectly possible that the rise in female offending is actually showing a rise in male sex offenders IDing as women.
For comment on sex offenders and other crimes, see The MoJ's Offender Management Statistics Bulletin, England and Wales
Sex offenders
The rise in the long determinate sentenced population is in line with the increasing number of sentenced sex offenders. As at 30 June 2018 there were 13,580 prisoners serving sentences for sexual offences, which represented 19% of the sentenced prison population.
The number of prisoners serving immediate custodial sentences for sexual offences is now at its highest level since at least 2002. This is consistent with the latest ONS 'Crime in England and Wales' bulletin which reports on the number of sexual offences recorded by the police in the year ending June 2018.
Full bulletin here: assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/729211/OMSQ-2018-Q1.pdf