What fuse to use?

Unfortunately it very much sounds like the work you have had done has not been done by an electrician! as no decent electrician would install a 3amp fuse on a flex for an outdoor socket! All outdoor sockets must be RCD protected and the work registered with your local building control dept under Part P of the building regulations. - I would imagine the DIY person who installed the socket has not registered the job, and probably does not even know what part P is all about. This leaves you as the householder open to a potential fine of up to £5000. If the work was done properly, you should have received an electrical certificate for the work? If you don't have one, then the alarm bells should be sounding and your best advised to get a registered electrician in and re-do the job properly. Check out the electricity safety council website for more information on RCD's and part P of the building regulations. http://www.esc.org.uk/public/guides-and-advice/electrical-safety-leaflets/

You may be OK to use a 13amp fuse. But really I would call the electrician that installed it to find out why he used a 3 amp fuse, probably just an error. I am suprised that an electrcian installed it using a plug in connection. Most electricians I know would have used a double pole switched fused spur. Plug in cabling looks very DIY.

AHHH hello, you should get your electrician back! why have you not phoned him? was he qualified?

13amp if the cable is a 1.5mm or bigger cable

bearing in mind that the outdoor socket has the potential for 26amps to be drawn, I would normally recommend a 13 amp fuse, but it depends on the size of the cable running between the pug inside, and the double socket outside, the fuse must always be rated lower than the cable ie, if the cable can only carry 10 amps safely, you fit a 10 amp fuse or less

all out door sockets should be protected by a 30mA RCD, get the sparky back to look at it and ask him for a certifcate and if he is regestared and to prove this to you if he is unable to do so stay clear and get a second person in to do, electric can kill so no room for error.