In he last day, three times I've seen a blackbird fly off when I walk out the front door from just behind my gate. I'm a little worried that it's starting to build a nest there with all the cats about, but surely it wouldn't?

I can't see any signs of a nest being built, but there are several bird droppings there so it must have been there regularly.

Its late in the season for building now, although blackbirds do find the worst places to build. I should think it more likely that there is a source of food near your gate that is attracting it.
We once had a pair nest on an upstairs window ledge completely exposed, which the magpies soon found and raided. If they had gone a couple of feet to the side they would have been hidden by a large tangle of ivy. That's blackbirds!

A pair nested in my garden last year right in a cat accessible place... The cats waited till the young were in the nest and then raided.. the parents were distraught for days afterwards in a big flap all over the garden. Luckily this year they must have found a better place as they have been much happier and I think a brood has been raised... But as Flowerpot says, they do choose the worst places!