Just a quick post to say that if your LO already has a passport, you can use a copy of your final adoption order to have a name change made in a new passport. No need (for this purpose, anyway) to worry about adoption certificates, etc.
Fast track service (you apply in person at a passport office) means you get the new passport back by post a week later. We recently did this and were so relieved - the passport advice line advisors weren't entirely sure it would work but said yes theoretically it should and I am happy to say it does!