The maximum legal power for amateurs in the U.K. is 400 watts peak and less than 150 watts average. In practice the average power will probably be less than 100 watts. Assuming that the aerial radiates equally in all directions (it doesn't, but more about that later), the 100 watts will be spread out over the surface of a sphere of radius 30 meters as far as the occupants of the house are concerned. That gives a power density of 100/4 x pi x 30 x 30 or about 0.009 watts per square meter. On the wall where I work there is a chart on which it is stated that the limit for public exposure is 0.2 milliwatts per square centimetre. 0.009 W/ sq. metre = 9mW/ sq. metre = 0.0009 mW/ sq. cm, so there is a safety factor of over 200. This will be more than enough to allow for the any directional properties the aerial might have.
    The situation in the shed is less clear, but 100W at one meter would be like 1W at 10cm (inverse square law and all that), so less worry than using a mobile 'phone, which wouldn't be 10cm from your head. As long as you use the shed only for storage and you don't stay in it for long periods there is probably nothing to worry about.
   It may be worth logging the times when the wi-fi misbehaves. The amateur also has to keep a log of the times when he starts and finishes transmitting. If you need to call in help, the post office will probaby refer you to Ofcom. Their inspector could compare the two logs to find out whether the problem is associated with the amateur's transmissions.
Domestic equipment has to withstand a field strength of 3 volts per metre to meet the relevant European standards. To convert watts per square meter to volts per metre, multiply by 377 and take the square root. 0.009W/sq. m is equivalent to about 1.8V/m, so if the aerial is directional (concentrates the power in a particular direction) there could be problems. As stated above, if the amateur is obeying the power limit, the onus is on the manufacturer of any affected equipment to solve the problem. If you are told "it must be okay because it meets the standard", point out that compliance with the standard gives a presumption of conformity with the requirements (allowing free movement of the goods within the European Union), not a guarantee of conformity.