*I’ve changed jobs twice in 2 years - both times I shows my passport as the right to work proof. No-one was after a birth cert or anything else.
And so far in my children’s existence no doctor or dentist or school or anything else has ever used middle names to ascertain their identity for routine stuff. It’s always name and DOB and address…*
OP. In the short term there will be little change using the DCs new names.
But the school will need to see documentation of the legal name (if you want to change it on their records), and will 100% need to see it if they want their new names on GCSE and A-Level entries and certificates. Universities need proof of names.
Longer term, every single job I've had in the last 15 years has required several forms of ID and proof of any name changes since birth. That's to work in a school. There are a whole range of jobs that will require this - and the official documentation to match - such as police, healthcare, law, social services, anything even remotely financial, local councils, education etc etc.
If your dc ever want to undertake any activity that involves working with children or vulnerable adults they will need a DBS - schools, brownie leader etc.
When applying for financial services they will need to provide official documentation.
When you apply for someone's death certificate you have to provide their name at birth.
This is a change that they will carry the burden of for a lifetime, in so many ways that you haven't even thought of. Your travel experiences are inconvenient but temporary.