There is a direct flight from Gatwick to Dubrovnik (I think it's once daily) by British Airways and Croatia Airways. It's not exactly cheap - around 300EUR round trip. I don't think that any of low cost carriers serve Zagreb at the moment (the closest airport with low-cost flights is Ljubljana in Slovenia, but I don't know whether there's a flight to London or somewhere near it). Other airports close to Dubrovnik are Split in Croatia, Mostar in Bosnia and Tivat, Budva and Podgorica in Montenegro, but I wouldn't expect you will find suitable flights there (except perhaps to Split). Another one is on island Brac, but I guess that once there you would just stay and enjoy swimming and windsurfing on Zlatni Rat beach in Bol
Dubrovnik is around 7 hours by road from Zagreb (at more or less legal speeds
)- about half of it on new motorway to Split, another half on old coastal road. Alternatively, one could take old road to Split and spend some time in cool
Plitvice Lakes national park, one of UNESCO Wold natural heritage sites.
Other means of reaching Dubrovnik from Zagreb is, again, plane (3-4 times a day, reserve well in advance) and bus (coaches are generally rather new and comfortable). The nearest places served by train are Split and Ploce (the later is served by international trains through Bosnia, used mostly by migrant workers "commuting" to Austria and Germany). Trains are rather slow and with somewhat lesser standard of comfort than typical European intercity lines.
Once in Dubrovnik, one can easily spend weeks without leaving the old city (except perhaps to hop to tiny island of Lokrum and its rocky beaches). A car would come handy for exploring
suroundings (Ston, Konavle, Cavtat, Peljesac, even Boka Kotorska, now in Monenegro), going to less crowded beaches and sampling offerings of many restaurants and taverns. (Many large rental agencies as well as myriad of locals operate here; since your visit will be past the top of the season, reservation might not be necessary; however, I would not try my luck if I were you.)
Obviously, you will want to take a tour of city walls (pick a not too hot day for that), explore tiny back streets and alleys, visit museums, climb to Fort Lovrijenac, see one of oldest (if not
the oldest) pharmacies still operating in its original location, churches, monasteries and synagogue, Rector's Palace etc. Regrettably, you will miss
Dubrovnik Summer Festival.
Regarding food and wine, things tend to change pretty quickly and I have not been there for several years, so it might be a good idea to take advice from a local and/or a recent guidebook. Expect good to excellent food (but there are tourist traps, too) and rather high prices. For example, "Nautica" restaurant outside of Pile (western city gates) was, last time when I visited, excellent, but one would expect to spend about 50 EUR per person without wine. Try "Rozario" on Prijeko, a street parallel to main Stradun (if it is still there) - it was my favourite way too many years ago...
Nightlife is supposed to be rather lively, but I don't have recent first-hand evidence
One idea for a day trip: Peljesac peninsula. Start with tour of Ston (another, smaller, walled city, with a wall going across the isthmus), then lunch in Mali Ston (fresh local oysters (
"kamenice"), mussel stew (
"dagnje na buzaru"), fish...). Then continue along the peninsula and visit some of small wineries and taste their Dingac (strong red wine; the
grape is a very close relative to Italian Primitivo and Californian Zinfandel) - some prior arrangement might be necessary. Finish visiting Orebic, a tiny town which gave many tall ship captains; some of their mansions are still there. (BTW, another walled town you will see across the strait is Korcula, purported home of Marco Polo; very worth a visit, too).
Have a very nice holydays!