Any organisation that falls within the public sector - government, schools, NHS, local authorities ... and any private or third sector organisations they commission to fulfill their public duties, is bound by the Public Sector Equality Duty which is part of the Equality Act. This says that they must have 'due regard to the need to' -
(a) eliminate discrimination, harassment, victimisation and any other conduct that is prohibited by or under this Act;
(b) advance equality of opportunity between persons who share a relevant protected characteristic and persons who do not share it;
(c) foster good relations between persons who share a relevant protected characteristic and persons who do not share it.
My understanding (IANAL) is that an equality impact assessment is not a legal requirement in itself, it's just evolved as the best way for organisations to show they're meeting this duty. So if they're not doing an impact assessment at all, how else are they addressing the PSED?
There have been some shockingly bad equality impact assessments, notably the Scottish GRA consultation (Annex N in the consultation paper - see pic.
I saw a worse one. The equality impact assessment for the Scottish census bill can be found here (save the link, it's bloody hard to find). Sex is lumped in under the heading 'Sex and gender reassignment' and then not mentioned again aside from consideration of how trans and NB people could answer the sex question.
The private sector do not have to meet the PSED unless contracted to carry out public functions.