Apostille vs. Embassy legalisation for United States
There is no consular stage on this route and there has not been one for a long time. A British paper takes one certificate from one office and is then finished. That makes the errand cheap to repeat, and repeating it is common: a family matter, a job, a study place and a property question can each call for a different sheet. The saving second time round is not in the fee but in the thinking.
United States's Hague status & what it means
The United States joined the Hague Apostille Convention in 1981, so nothing about the arrangement is recent or provisional. A certificate issued today is read in the same way as one issued decades ago. Stability of that kind is worth leaning on when you plan, because the steps you write down after a first run will still describe the route accurately when you come back to them later.
The Liverpool → United States route step by step
Liverpool Register Office at St George's Hall, St George's Place, Liverpool L1 1JJ issues certified copies of births, marriages and deaths, so a civil record can be reordered rather than risked. The FCDO Legalisation Office then adds the certificate on behalf of the whole United Kingdom, working entirely by post with no counter to attend. Two offices, one charge for each item, and nothing waiting after the second stage.
Cost & timing for this route
The published figures are £45 / document by standard post and £40 / document on the next-day service open to registered businesses. Both are counted per item, so a note of what you paid last time is also a note of what the next run will cost. Turnaround is not published as a fixed number, which is exactly why the date the last batch went off is the most useful line in your own record.