I read through the letter. It looks like performative lawyering, aka playing the gallery.
If you boil it down, it says "we think you're wrong, withdraw the interim advice etc. or else."
The "or else" is the important bit, and there isn't much of it. The letter mentions: "The Interim Update is thereby an unlawful statement of policy or guidance applying the principles in R (on the application of A) v Secretary of State for the Home Department) [2021] UKSC 37, [2021] 1 WLR 3931 and Gillick v West Norfolk and Wisbech Area Health Authority [1986] AC 112. The Claimants will seek a declaration to that effect. "
Both of those cases are easily distinguishable from this situation.
Other than that - the phrase "judicial review" is mentioned, but only on the first page. Unless I'm mistaken nowhere does it actually say GLP will ask for a judicial review.
In any case a judicial review is applicable to a government decision, which this "interim update" isn't. It's a claim that a decision (which this, again, isn't) hasn't been reached by the correct process. A judicial review isn't appropriate if someone thinks a decision is wrong. But in any case there's no decision here to review. I suspect that GLP knows that.
On the subject of codes of practice:
The EHRC is a creature of statute, created by the Equality Act 2006. Under that Act, the Commission can, or in some cases must, issue codes of practice (s.14).
A code of practice shall shall contain provision designed—
(a)to ensure or facilitate compliance with the Equality Act 2010 or an enactment made under that Act or, (b)to promote equality of opportunity.
...
"(6)Before issuing a code under this section the Commission shall—
(a)publish proposals, and (b)consult such persons as it thinks appropriate."
Note: the choice of whom to consult with is entirely at the discretion of the Commission.
Before issuing a code, the Commission has to submit a draft to the Home Secretary. If the Home Secretary approves the draft, they lay a copy before Parliament which has 40 days to pass a resolution disapproving the draft and if that doesn't happen the code goes into force.
Once that happens it has the imprimatur of Parliament. A court will not overturn something that was approved by Parliament.