Venus has been closely studied for a long time. In fact in 1961 it was the second object (the moon was the first) to be radar mapped from Earth. Also in 1961 the Soviets sent a probe that did a flyby but due to a malfunction it wasn't until 1962 when the American Mariner 2 spacecraft, which flew past Venus. A modified Ranger Moon probe, it established that Venus has practically no intrinsic magnetic field and measured the planet's temperature range as 490 to 590 K.
The initial environmental measurements of the surface temperature of Venus is what sparked the idea of "Greenhouse Gases" because the surface was much hotter than expected based on solar input.
The first successful landing on Venus was by Venera 7 on December 15, 1970. It remained in contact with Earth for 23 minutes, relaying surface temperatures of 455 °C to 475 °C (855 °F to 885 °F). Venera 8 landed on July 22, 1972. In addition to pressure and temperature profiles, a photometer showed that the clouds of Venus formed a layer, ending over 22 miles above the surface.
Since then we have continued to measure and observe with a total of about 26 successful probes, however we don't do any sort of weather predicting similar to the extreme detail done on Earth. The study of climate on other planets is done mostly by observing what is happening and then seeing if our models allow this to happen. If not, then we have to figure out what was wrong with out understanding of that planet's mechanisms.
Currently the Venus Express is a long term probe that has been sending back data since 2005. It has led to deeper understanding of the atmospheric dynamics that drives climates like Venus and Earth. In addition Earth based observations like HIPWAC help to monitor the climates of other planets to better understand how they work.
When I get more time, I'll update this with some modeling information.
Edit: Some of the more challenging aspects of Venus is the extreme thermal conditions. This is the same engine that drives all the significant weather on Earth as well. Winds on Venus travel at about 224 miles per hour and the velocity changing with altitude. Cloud wind speeds decrease at higher elevations, and the highest clouds are estimated to be traveling at a few miles per hour. The cause of cloud movement is still being studied at NASA. These clouds produce lightning bursts that are the result of sulfuric acid in Venus's atmosphere. The acid produces a constant electrical charge that is discharged to the planet's surface as lightning.
The Venus Express took a mosaic of images that has been helpful in simulating the extreme vortex type weather that has been observed.
Simulations of vortex models have achieved some likeness to observations made of Venus.
What institution/s have, thus far, developed/attempted to develop a meteorological model of Venus?
Some Venus global atmosphere models have been built by the CCSR/NIES (Center for Climate System Research, University of Tokyo; National Institute for Environmental Studies, Japan) GCM and the United Kingdom Meteorological Office (UKMO) Unified Model (UM).
You can even get an open souce planet weather simulator that uses parts of these models for scientific research.
At what stage of development are such models of Venusian climate?
Are these models capable of generating a tentative map of Venerean climate several weeks into the future?
Due to the lack of ground and atmospheric sensors Venusian weathermen are quite inaccurate. The stage of their models are just in the "can we get a simulation to match what we see on a planetary scale". There is nothing done on a micro scale such as we are used to here on Earth. No surface wind forecasts, storms, daily weekly forecasts. We are just trying to emulate some of the global aspects that we observe.
On Earth we have many sensors all over the planet and in orbit to calibrate and check our models against, without that real-time feedback there is no way to refine such models. So trying to model micro weather on Venus without being able to verify it is mostly a waste of time.
Towards Understanding the Climate of Venus: Applications of Terrestrial Models to Our Sister Planets.