A PCBA is usually a component, not a finished product, hence outside the scope of CE marking. I believe originally when CE marking came in, it was illegal to CE mark something that wasn't in scope, but ISTR this subsequently changed
Problem is, with WEEE and RoHS basically everything with electronics got into the scope of marking.
Anyway, what has to be marked is what the end-user gets.
So if your PCB is something the end-user really can hold in his hands, because he could order that bare PCB, you'd need to think about (the applicable parts of) CE marking.
This means, if you have a product that allows for user-upgradable-options, you may have CE marks all over the place

If I'm not mistaking, you're from the UK, so you can't really do a CE self-declaration... it's the same we EU guys can't do UKCA or anyone of us could do FCC SDoC.
This needs to get signed by someone who could get charged in case something's really problematic enough to bother someone.
If your customer uses your design in his product, he's the one who needs to do the CE stuff... but that's not really a certification... it's "just" a declaration that tells authorities that you think you have done everything needed.
In theory, you could go to some notified body to do certification... but nobody really wants to do this because of the troubles you're gonna get into.
A notified body really needs some test for every aspect.
You may just refer to some components datasheet for RoHS when you purchase stuff from reputable manufactureres... a notified body will need some test-report to accept this.
So in your case, I'd review your contract...
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