Compute an asymptotic interval with an approximation (Delta method)

Dear all,
I am performing an analysis where I need to calculate the percent attenuation between two CI (of OR or RR or HR)

Essentially, I have a CI for an unadjusted model and a CI for an adjusted model, and I want to compute the attenuation percentage between them, due to the covariable in the adjusted model. I'm doing this for a SR + Meta analysis, therefore, I do not have any data, I only the CIs from the articles.

The percentage attenuation is calculated as following:

percent_att=100*(Beta_unadj - Beta_adj) / Beta_unadj //where Beta = log(HR) or log(OR) ...

Now, I also want to compute the 95%CI for percent_att (which would be my parameter of interest for the MA).

I was told by a statistician that I could do that with the following method: "Asymptotic Interval with an approximation (Method of the Delta, Wald IC)". I checked different methods (on the internet) but I don't have the background (Biology...) to understand this nor set up formulas for lower and upper limits.

Therefore, if someone would help me with proposing formulas for lower and upper limits, I would be extremely grateful.

As mentioned above, I have the 2 CI (unadjusted and adjusted) as well as the number of participants in each study.

Thank you very much for your kind help.