It's worth having a look at Stonewall's accounts here:
register-of-charities.charitycommission.gov.uk/charity-search/-/charity-details/3992465/accounts-and-annual-returns
Latest figures are for 2019 (details p.28 onwards).
They receive grants from the DfE, FCO, and Scottish and Welsh Governments. Total income from government sources in 2019 was £702,295.
Fees received from membership of programmes such as the Diversity Champions programme and Schools Champions, plus speaker fees and 'consultancy requests': £3,269,477. Fees from other 'training': £633,954.
Much of this is also public money. Sex Matters estimate that about 250 Government departments and public bodies are members of Stonewall schemes. If they pay around £3,000 p.a. each for membership, this is around £750,000, plus any additional fees paid for extras such as 'training'. There is also income from state schools.
Nearly half of Stonewall's income (about £8.3m in 2019) comes from their Champions programmes and training. They stand to lose a lot if organisations pull out of these schemes.