The lesser known superviosr has a grant for you and it is from the EPSRC, am I right?
In that case, here's my input for you, if it is not too late or else for anybody else.
The EPSRC grants are very competitive, not for the student but for the professor to get hold of. First if only he has a decent publication record and popular among other scientists, he could submit a good proposal to the EPSRC with an innovative idea.The EPSRC then would send it for peer reviewing to three independent senior professors in the field. Unless the reviewers agree almost unanimously,the proposal won't be a success. So in this case, the lesser known professor should have done a good job and the Oxford professor might well have refereed his proposal and approved it.It'd be my pick.