Just a quick update to let you know how it went. I always appreciate it when posters reply to their threads with info on what really happened.
We arrived into Nairobi on time at around 4pm. Despite wanting to get moving, we were nearly last off the plane (sat at the back and it also contained a presidential delegation from Comoros - at least we got a red carpet!), but it didn't matter because everyone had to get into a shuttle bus to be taken to the terminal building. We zoomed to the immigration desks and were second in the queue. Visa forms and landing cards were handed out on the plane, so we'd filled them in in advance. We paid for our transit visa in US$ and were dealt with in just a few minutes. After that, we were free to leave the airport. We'd checked our hold baggage through to London so didn't need to collect it - we just had hand luggage with us.
At the front of the airport, we went to one of the tourist info desks and asked if there was time to get to the National Park. We were told there was, so hired a driver to take us there. We were on the road by 4.45pm, so I thought we made good time. The driver took us to the East entrance to the park, reasoning that traffic was too bad to get to any other entrance. (We saw the terrible traffic jams for ourselves later.) It was about a 20 minute drive away. For some reason, it wasn't possible to pay for tickets at that entrance, but they let us in and the driver promised to head straight for the main entrance to pay. He did this, but it meant that we were in the park before it started to get dark, which allowed us to see a few animals along the way. If we'd taken the road outside the park, we'd have never got there in time. After crossing to the main entrance and paying (the driver took care of this) we set off back into the park to look for more wildlife.
It was a great little safari - we were lucky enough to see ostriches, antelope, giraffe, baboons, lions, zebra, rhinos, buffalo, lots of birds... everything we'd hoped to see. We stayed until after dark, leaving about 6.45pm and were well satisfied.
We ended up going to Carnivore afterwards - the guy at the tourist desk said there would be vegetarian food on offer for my partner (they had a few choices) and I wanted to experience the place for myself. And it certainly was an experience. So much meat! We had a good meal there and then returned to the airport stuffed but in plenty of time for our next flight.
Things worked out very well and I have to say it was the best layover I've ever had. From the look of Nairobi airport, I'm certain we made the right decision to get out of there! Thanks for the advice everyone.