I am a commercial electrician in the US and I'm not very familiar with UK standards or code, but I should be able to help a little.
From what I understand, a typical household runs lighting on a 5A fused circuit. Watts = Volts*Amps = 230*5 = 1150 Watts maximum total on the lighting circuit. It the US we us a maximum of 80% of the rated load, which would equate to 920 Watts. This allows for voltage spikes when first turning on and off the lights, random power surges, etc. Keep in mind that this rating is for the entire circuit, so if there are any other lights, receptacles, or appliances attached to the same circuit, you would have to subtract those to figure out how much power you have left to use for your project.
The wattage of each light bulb is going to be your biggest factor. If each bulb was 100 watts, multiplying that by 15, you are 1500 watts, 580 watts over your 100% fuse limit, and that's if there is nothing else at all on the same circuit (unlikely). If you used 40 watt bulbs, you are at a total 0f 600 watts, which leaves you with room to spare, and would probably be plenty of light for a vanity mirror like what you have in mind.
As far as the actual construction goes, running mains power through a wood frame to your lampholders with multiple splices isn't the safest diy project, but it can be done safely if you know what you're doing. Maybe somebody else can comment on that issue.
I would use low voltage LED lighting if it were my project. It's safer and uses much less power, but would probably be more expensive to put together and it would involve some more design considerations.
Good luck on your project and keep us posted.