Bowe moves into the midfield from the right wing to join another newcomer, England's Riki Flutey, who takes over from Jamie Roberts, whose sprained wrist sustained in the second half of last weekend's second Test in Pretoria hasn't fully healed.
Because of Bowe's switch, McGeechan has introduced a new wing pair. Ugo Monye returns to Test action on the right and Shane Williams takes over from Luke Fitzgerald on the left to make it four changes behind the scrum from last weekend.
Up front, Vickery returns for his fifth Lions Test and Sheridan makes his first Test start in the front row. The second row remains the same, but the back row has been revamped following the 28-25 defeat at Loftus Versfeld.
Irish No8 Jamie Heaslip remains the one constant throughout the three Tests, while Wales flanker Martyn Williams gets his first Test start on his third Lions tour. He switches with David Wallace, who drops to the bench, while 30-year-old Joe Worsley replaces fellow Englishman Tom Croft on the blindside of the scrum.
Croft drops to the replacements bench, where the Lions have employed a 5-2 split between forwards and backs. Ross Ford, Alun Wyn Jones and Harry Ellis are retained from last weekend's bench, while Irish tight-head prop John Hayes, who flew out as a replacement for the injured Euan Murray, provides cover for the front row.
Wallace switches with Martyn Williams and James Hook is included in the Test 22 for the first time.
British & Irish Lions (vs South Africa, 3rd Test)
15 Rob Kearney (Leinster / Ireland)
14 Ugo Monye (Harlequins / England)
13 Tommy Bowe (Ospreys / Ireland)
12 Riki Flutey (London Wasps / England)
11 Shane Williams (Ospreys / Wales)
9 Mike Phillips (Ospreys / Wales)
8 Jamie Heaslip (Leinster / Ireland)
7 Martyn Williams (Cardiff Blues / Wales)
6 Joe Worsley (London Wasps / England)
5 Paul O'Connell (Munster / Ireland, Captain)
4 Simon Shaw (London Wasps / England)
3 Phil Vickery (London Wasps / England)
2 Matthew Rees (Scarlets / Wales)
1 Andrew Sheridan (Sale Sharks / England)
16 Ross Ford (Edinburgh / Scotland)
17 John Hayes (Munster / Ireland)
18 Alun-Wyn Jones (Ospreys / Wales)
19 David Wallace (Munster / Ireland)
20 Tom Croft (Leicester Tigers / England)
21 Harry Ellis (Leicester Tigers / England)
22 James Hook (Ospreys / Wales)