I travelled to Jordan on business with my UK passport and it was stamped on arrival. Will I have challenges later going to Israel or if I can go to Israel with the Jordanian stamp, will I have challenges returning to Jordan at a later date?
Yes, you can enter Israel or Jordan after visiting the other without problems. The two signed a peace treaty in 1994 and have permitted travel to each other ever since.
Of course, you may be questioned in Israel about what you were doing in Jordan, but if you went there for legitimate business reasons, you'll be fine.
Yes, you can enter Israel, but be prepared for the possibility of getting grilled hardcore at Israeli immigration for having been to Jordan (they can be even worse than the US at times - I experienced it even though I am Swedish and have never been to an Arab country).
Not saying it will happen, but it could. Bring as much documentation as possible proving the purpose of your Jordan visit
Entering Jordan after Israel is 100% fine, and not just due to the fact that Israel doesn't stamp passports anymore (at Ben Gurion at least).
You can enter Israel directly from Jordan. There are three open border crossings and I myself crossed into Eilat, Israel from Aqaba, Jordan at the Yitzhak Rabin crossing between the two sister cities. Obviously, being a European, I had a Jordanian visa, entry and exit stamp in my passport. If the direct crossing is possible, entering Israel at a later date is possible, too.
When having entered Jordan, about three-quarters of the group I was with had been to Israel three years prior. Nobody had any issues entering Jordan whatsoever. Furthermore, our guide — a professor of theology at the university of Innsbruck — semi-regularly visits both these countries on a single passport.
We (i.e. mainly the guide, but a little cross-checking was performed) were asked quite a few questions about what we were doing in Jordan. I’ll imagine, if you turn up at Tel Aviv airport immigration at a later date than just after exiting Jordan, they will have more. It’ll probably raise a questioning flag for them but not necessarily a red one.
Israel does not stamp passports a lot anymore. Tel Aviv airport seems to be fully equipped for only supplying you pieces of paper instead of passport stamps. When I crossed the border into Israel at Eilat, I got a stamp but I have since read here that others (who did the trip at an earlier date) received a paper instead. I’m pretty sure you can specifically request a paper rather than a stamp at each port of entry.