These are the closest approximations I could come up with without resorting to the phonetic alphabet. I’ve broken the words up into syllables (not always logically) and respelled them as if they were British English words. I’ve also given the meanings of the placenames.

magyar mod-yor Hungarian

Bükk book beech

Hernád hair-nahd

Kosice kohsh-eats-eh

Lillafüred leel-loff-oor-ed lily spa

Miskolc meesh-cohlts

Palota-szálló pol-oh-ta sahl-oh palace hotel

Sajó shoy-oh

Tapolca top-ohlt-sa

Tisza teess-sa

Aggtelek og-tel-eck old land

Bánkút bahn-coot lord’s well

Bodzás-rét bohd-zahsh rate elderflower meadow

Bükkszentkereszt book-sent-kerr-est holy cross

Galuzsnya-hegy gol-oozh-nee-a hedge dumpling hill (?)

Hollós-tető holl-ohsh tet-ur raven top

Királyasztal kee-rye ost-oll king’s table

Kurta-bérc koor-ta bayrts short hill

Létrás-tető lay-trahsh tet-ur ladder top

Lófő-tisztás low-fur tee-stahsh horsehead clearing

Nagymező nodge-mez-ur great meadow

Szelcepuszta selt-sep-ooss-ta ? plain

Szép-hegy sayp hedge pretty hill

Szögliget sug-lee-get nail grove

Vivrát-hegy veev-raht hedge ? hill

Áron Less ah-rohn lesh

János Sőtér yah-nohsh shirt-airr

László Solyom lah-sloh shoh-yom

Száva Zsigmond sah-va zheeg-mond

Tibor Zsigmond tee-borr zheeg-mond

Zsolt Gerzsényi zhohlt gair-zhayn-yee

cél tsayl finish

rajt royt start

tájfuto tie-foo-toh orienteer

tájfutás tie-foo-tahsh orienteering

világbajnokság vee-lahg-boy-nock-shahg world championships

Just discovered these notes Dave Peel made a year ago. (pdf)