Sorry for the delay in replying.
I ended up booking a hotel car for around £40 for the day with a driver. We drove straight to Lekki (IIRC around £4 entrance) which took around 90 minutes from near the airport. You have to join a tour, and there was a half hour or so wait. There's a small depressing cafe (bring your own snacks!), and a few information boards around the centre about the flora and fauna. The tour was more an employee leading a group of tourists (a few locals, a few expat families, a few Nigerians living in the UK visiting family) along the walkway than doing any active guiding.
The walk around the centre was FAR better than I had hoped for (as i'd done little research on what the conservation centre actually involved), with long sections over a swampy area filled with monkeys, before getting to Africa's longest suspended walkway through the rain forest (you have to pay a little extra to do this). To walk all the trails took around 45-60 minutes depending on your pace (the guide vanished after a while, so we were free to wander).
Other than monkeys, we didn't see a whole lot of other animals, but we were happy to just enjoy the walk and sights and smells of the forest.
On the way back, we asked to stop by the beach (which looked very close to the main road). Bit of an error as the 300m section to the coast took around 20 minutes of dodging potholes, and the beach when we arrived was not the paradise we were hoping for! Anyway, we got a few photos and headed back to the hotel, via KFC for an very non-local lunch!
We considered going into the city to have a look around as we had the car and driver for the whole day, but decided to play it safe and head back to the hotel, rather than risk getting stuck in rush hour traffic for the sake of an Instagram photo. Overall it was a great excursion, and much more enjoyable than sitting by the hotel pool waiting for a midnight flight.
I was a little worried about security before I went, with this being my first time in Western Africa. Now that I'm back, i'm pretty reassured that it's fairly safe - no armed guards necessary! I'm now planning a little trip to Benin for September - any advice for that?!
Thanks again for the input from you all!