Part P regs saying it was BC's responsibility to perform these tests.
Not quite it says where the installers are qualified to carry out inspection and testing and completing the appropriate BS7671 installation certificate, they should do so. (1.22)
1.23 c. Where installers are not qualified.
Then it is up to building control to decide (1.25) and if they do then they pay for tests.
However where a non qualified person wants to do electrical work they (LABC) must of course have the ability to say no. Otherwise how could they become responsible for making sure the work is safe. (1.24)
They do not have magic wands and there must be people who have no idea what they are doing and the only way they could safely do the work would be for the LABC guy to watch and instruct them in every step which of course they can't do.
Also any inspection and testing they do will not be passed to the DIY person so although he would have a completion certificate he would not have a Minor Works or an Installation Certificate.
What the document says is if the LABC does not believe your results and does there own they can't charge you.
1.27 states quite clearly that LABC will not issue an installation inspection and test certificate.
How the LABC can prove if anything went wrong that they fulfilled their responsibility ensure the work is safe and compiles with all relevant requirements in the Building Regulations (1.24) without inspecting and testing I don't know.
In the same way as I know of no court case where some one has not claimed to be a member of a scheme and has completed work which complies with BS7671 neither do I know of any case where the LABC has been taken to court for not for-filling their duties under Part P.
They have been taken to court for Baby P. And people have been taken to court for substandard work and for claiming to be a member of a scheme when they were not.
And really speaking unless LABC can be prosecuted for not doing their duty then the whole idea of Part P must be nothing more than a tax for improving ones own home.