Tokyo to Kagoshima is ~ 7 hrs via train on non-Nozomi. The nationwide JR pass isn't valid for Nozomi so you can't jump on all shinkansen. There are enough slightly slower shinkansen & express trains to make the routing possible, though, if you're not in a hurry and you can do it same day if you leave by 4:03 pm from TYO. You can still reserve a seat via the ticket office if there is an available seat on the day of travel. The NRT to Kagoshima train fare would definitely make the JR pass worth the fee but you might want to see the price difference for an open jaw airfare ticket from UK-NRT-KOJ if the schedule lines up. Also take a look at UK-NRT, HND-KOJ and the tourist airfare deals if you've already booked the r/t UK flights https://www.japan-guide.com/e/e2364.html.

You can compare the price of 1 way airfare to Kagoshima & 1 way train between Kagoshima, Kumamoto, Osaka, Tokyo vs all rail on HyperDia ???? ???? ??? ???? ??? ???????