Well, obviously the Soviets could do better. If the Chinese get theirs started in the '70s (historically there were several attempts by interested Chinese figures to start a human space program, but all except Shenzhou failed due to the poor cost:rewards ratio and the limited technology and industrial base), and Europe decides to go ahead with Hermes, then that makes 4 by the year 2000. Going into the future, India could have a human space program soon, or they could have started on it sooner, thereby getting them in under the deadline. Brazil has a good location, and perhaps if it had been luckier it could have had a space program of its own, even with human flights. Japan could obviously do it if they wanted to. Iran is somewhat plausible, but does have some difficulties depending on the political situation. Israel would be tricky, but they probably have enough technical and economic strength to launch a few people into orbit, though much more would be very difficult. So there's 9 plausible programs that could have gotten off the ground (err...so to speak) by 2010. Obviously an easy trick to boost the numbers is to split up the European programs into component national agencies, but then you face bigger questions about funding and will than with the multi-national agency, particularly with lunar and Mars flights.
Now, as for going to the Moon, the only really plausible candidates are the US, USSR, and the ESA. Japan runs a close fourth, and China is a more distant fifth. India is just behind China, and Brazil, Iran, and Israel are very unlikely. The problem with the bottom 5 is they start in a worse place technologically and economically, and going to the Moon is tricky and expensive. To get Brazil to the Moon, for instance, you would really need PoDs in the 19th century to make it industrialize faster. Then it would probably be able to compete. Japan gets rich enough and starts off advanced enough that it can maybe keep up if it tries hard enough, but China and India start in a very poor position, so they'll have a tricky time of it. Obviously Brazil, Iran, and Israel are in an even poorer position than China or India due to their small comparative size and relatively poor development (for the first two, at least).
The only plausible candidates for Mars are the US and USSR, with maybe the ESA if they decide to get on it. Of those, the US is by far the most likely to be successful at getting there.
EDIT: The trick to getting more programs going is probably to have the superpowers less willing to fly non-national astronauts on their spacecraft (in particular the USSR, which saw that as a propaganda trick). That means that countries that want astronauts have to launch them themselves. Of course, that's going to be difficult...