There are several places in central london where you can play golf indoors on simulators with a full swing and real clubs and balls, including:

Urban Golf

Urban Golf Kensington
Kensington Shopping Mall
125 Kensington High Street
London
W8 5SF
Tel. 020 7248 8600

Urban Golf Smithfield
12 Smithfield Street
London
EC1A 9LA
Tel. 020 7248 8600

Urban Golf Soho
33 Great Pulteney Street
London
W1F 9NW
Tel: 020 7248 8600

Metro Golf

Metro Golf Academy
19 Sheldon Square
Paddington Central
London
W2 6EP
Tel: 020 7266 7950

City Golf and Health Club

40 Coleman Street
London
EC2R 5EH
Tel: 020 7796 5960

City Bunker

Cannon Workshops
Hertsmere Road
Canary Wharf
London
E14 4AS
Tel: 020 7537 7940

Cannon Golf

2 Cannon Mews
98-102 Sydenham Road
Sydenham
London
SE26 5JX
Tel: 020 8659 1708