Hi,
Kind of depends where you regard as "Central London" but Ealing Golf Range is just north of the A40 fairly close to the short, flat, compact Northolt Golf Course. A bit nearer the centre but still close to the A40 is the flat shortish 9 hole Perivale Golf Course. The 9 hole (8 par 3's) Horsenden Hil course (excellent views) is about a mile north of Perivale G.C.
Duke's Meadow Golf Course and range may meet your requirements. It is next to Chiswick Bridge and is a well maintained 9 hole par 3 course.
Lee Valley Golf Course and range may be a bit far out. 18 holes but short.
Northwick Park range and course out near Harrow is a stunningly well maintained newish facility. Was 6 holes, now nine and getting cheaper by the second!
Central London Golf Centre near Wandsworth Common has a short range (play into netting) and a short, flat, compact (as in crash helmets) 9 holes.
Out east, possibly the far east, Crowlands Heath, which is beyond Ilford! looks a good bet.
You will have to look at a map to see the relationship of public transport to the above and there may be other facilities of which I know nothing but hope this helps.
