You are remembering some of the series of adventure novels about the character Richard Blade, published under the house name of "Jeffrey Lord." I've read a few of them.
According to Wikipedia, the English-language version of the series lasted for 37 volumes, #1 being published in 1969 and #37 in 1984. The first eight were actually written by Manning Lee Stokes. Then Roland J. Green took over, writing nearly all of the others -- except that #30 was written by Ray Faraday Nelson. (Don't ask me why.)
Complicating the issue: There were two sets of sequels which were each published in a different language.
After the first six volumes were translated into Russian, a bunch of sequels were written and published in Russian. As far as I can tell, those never had English translations.
Meanwhile, another set of sequels was coming out in France, written in French by various authors, and that version of the Richard Blade series made it all the way up to #206, that last one being published in 2012. (Again, I don't see any indication that the French sequels were ever published in English-language editions.) Before today, I only knew about the English-language series, which began with The Bronze Axe in 1969, establishing the basic premise much along the lines you remembered.
I thought of Richard Blade as soon as I saw your question, but what really cinches it for me is your description of the adventure which ended with his coming home with a brilliant dragon-creating geneticist in tow. I will now offer a few quotations from that book to demonstrate that it must be the same one you remembered.
The book in question is The Dragons of Englor, #24 of the original English-language series, and written by Roland J. Green. Used paperback copies are available on Amazon if you follow the link, but it appears there's not yet been a Kindle e-book edition.
Here's a summary of the basic premise of the series. It comes from the first chapter of The Dragons of Englor. Obviously provided for the benefit of any new readers.
He was the only living human being who could travel into other
Dimensions and return safely. It was because of Blade's uniqueness
that he and J were walking along the echoing corridor far below the
Tower of London. At the end of the corridor lay a series of rooms, and
in the last of those rooms stood an enormous computer. That computer
was the creation of Lord Leighton, who had the most brilliant mind and
usually the worst temper among all of Britain's scientists. Richard
Blade's brain would be linked to that computer, so that they formed a
single circuit. Then Lord Leighton would pull a red master switch,
activating that circuit, and Richard Blade would whirl off
They called that "somewhere else" Dimension X. When the great computer
had finished twisting Blade's brain and senses, he saw and smelled
somewhere else, heard and felt somewhere else, fought and moved
somewhere else. Somehow he always survived and came back alive, sane,
and reasonably healthy, to tell of what he had done and seen in the
unknown. He was the only living person who could do that, in spite of
all the efforts made to find others.
And here's a bit from the final pages of the novel, in which Richard Blade is being brought up to date by J (his handler from MI6) regarding the practical results of his latest mission. He's been sent to a parallel world where the local version of England is "Englor," and the local version of Russia is creatively called "Russland." As you remembered, the latter has made breakthroughs in genetic engineering, enabing them to produce dragons as terrifying weapons of war. Rilla Haran, a young woman, was the Russlander geneticist who had a lot to do with making the crucial discoveries, and she is so disgusted by how her work has been used for warfare that she wants to defect to Englor.
Richard Blade helps her do so, and in the process he obtains copies of her detailed notes on what she had discovered in her laboratory work. Then, although this part had not been planned in advance, he just happens to be holding her hand tightly when he feels the strange sensations that mean he is about to be teleported back home to his native England. (He has no control over the timing of this process.) Rilla ends up being brought along for the ride. But, as mentioned in the part I quoted above, other human beings just don't have what it takes to cope with these interdimensional jumps the way Blade can. I'm not sure, but I get the impression that this may have been the first time in the series that he actually "carried a passenger" all the way from their native continuum to ours.
So here's the excerpt that tells us how it all turned out, from MI6's point of view:
"What about Rilla?"
"Her notes are exceptionally complete, by the standards of her own
Dimension. However, much of what was common knowledge there isn’t
quite so common here. Again, we have something whose value is enormous
and can be realized fairly easily. It won’t be another case like
teksin. But it will be a few years before we can use Miss Haran’s
discoveries, either for curing cancer or for building dragons."
The attempted humor fell flat. Blade sensed that J’s heart was not in
it in any case.
"No, I meant -- how is Rilla herself? I haven’t been let in to see
her, so I assume she’s still recovering from the transition, but--"
"Richard," said J quietly, and the soft voice held enormous compassion
for the younger man. "Rilla has quite recovered, physically. But
mentally -- she is not doing too well."
"How -- badly?" said Blade.
"She has no more mind than a six-month-old baby," said J.
There was a long silence. Blade stared into the fire. He had seldom
felt worse in all his life in any Dimension. Rilla’s mind was gone,
and when all was said and done, it was his fault. He could have left
her in Englor.
I don't think there's any question that this is what you once read. "Dragons of Englor" could easily be remembered, decades later, as "Dragons Over England."