  Studies of transiting Neptune-size planets orbiting close to nearby bright
stars can inform theories of planet formation because mass and radius and
therefore mean density can be accurately estimated and compared with interior
models. The distribution of such planets with stellar mass and orbital period
relative to their Jovian-mass counterparts can test scenarios of orbital
migration, and whether "hot" (period < 10d) Neptunes evolved from "hot"
Jupiters as a result of mass loss. We searched 1763 late K and early M dwarf
stars for transiting Neptunes by analyzing photometry from the Wide Angle
Search for Planets and obtaining high-precision ($<10^{-3}$) follow-up
photometry of stars with candidate transit signals. One star in our sample (GJ
436) hosts a previously reported hot Neptune. We identified 92 candidate
signals among 80 other stars and carried out 148 observations of predicted
candidate transits with 1-2 m telescopes. Data on 70 WASP signals rules out
transits for 39 of them; 28 other signals are ambiguous and/or require more
data. Three systems have transit-like events in follow-up photometry and we
plan additional follow-up observations. On the basis of no confirmed detections
in our survey, we place an upper limit of 10.25% on the occurrence of hot
Neptunes around late K and early M dwarfs (95% confidence). A single confirmed
detection would translate to an occurrence of 5.3$\pm$4.4%. The latter figure
is similar to that from Doppler surveys, suggesting that GJ 436b may be the
only transiting hot Neptune in our sample. Our analysis of Kepler data for
similar but more distant late-type dwarfs yields an occurrence of
0.32$\pm$0.21%. Depending on which occurrence is applicable, we estimate that
the Next Generation Transit Survey will discover either ~60 or ~1000 hot
Neptunes around late K and early M-type dwarfs.
