I would expect that this nomination is entirely at CDS' own wish. I'd also expect that he has been encouraged to put his hat in the ring, having made something of an impression on his fellow Chiefs of Defence, and that there will have been a good deal of background work with other nations in support of the nomination. The reported US backing is potentially significant and suggests to me that a number of others feel it's time for one of the 'larger' NATO nations to fill the post again. (By custom, and as it fills the Major NATO Commander posts, the USA does not nominate for CMC, but the Deputy CMC is always American.)
So, I don't see it in any way as his possibly being 'kicked upstairs' by the UK MOD, and I would have thought that to be elected by one's peers to such a post would be a significant end to an already successful career. He remains CDS until moving to HQ NATO, if elected, or until his national retirement, so I would not see him as in any way 'fatally wounded' should he not be elected. His potential replacements as UK CDS are, as ever, the current Single Service chiefs and VCDS.
The matter of DSACEUR, until now a UK post, and mentioned in the Times article, should be another matter altogether - and I would earnestly hope that Brexit is not allowed to cloud the NATO picture. But time will tell on that.