Dibies,
You were an SFA student and as such subject to the SFA requirement for all logbook entries to be signed by your FI. Additionally, you trained in the U.S.A and it is a requirement for all lessons to be certified by the instructor following the flight - part of the endorsement process under FAR61.87. Oxford issue all integrated students with their own logbook which is an AFE JAR-FCL one with OAA branding. I have no doubt they would be prepared to sell you one. It is quite smart and certainly fit for purpose. I mentioned it in another thread and am still curious though - if you have committed to modular training with an expensive FTO why are you asking these questions on a public forum? Would it not be simpler just to pick up the phone to Kidlington or drop them an e-mail?
I wish you good luck - I hope your training at CHD has helped you get on top of things with respect to your landings.