"He just sort of popped up and got as much of a fright as we did," he said.
"I was pretty happy, pretty excited. Elation really, a lot of elation. I gave him a big cuddle and something to eat and drink.
"Out in the tussock we came across him and he was getting blown sideways. Up on the top there, it was ugly. The visibility wasn't too bad. We were in the cloud line. But the wind up there was amazing."
Thames was in good condition, Mr Wakefield said, despite soreness in a back foot, although "you wouldn't know he's been out running round in the bush for a week".
Television crews and newspaper reporters and photographers had greeted Mr Wakefield and Thames alongside fellow Wairarapa dog handler Constable Lloyd McKay and other handlers including officer in charge of the Wellington police dog section, Senior Sergeant Mark Davidson.
"The support's been huge. That's what's kept us going. The guys that've given up their time is just phenomenal. LandSar guys have taken time off work - they've come up on the weekend and that's why we're here.
"Realistically I wouldn't have been up there by myself in the conditions - they were pretty ugly up there but those guys really know what they're doing."
Mr Davidson said Mr Matheson had been pivotal in resuming the search on Saturday and ex-deer culler John McCann, who spotted paw prints in heavy weather, helping bring the last-ditch operation to a successful end.