The fact is, if you tried to build a bike from scratch with no 'bits box' or prior components, unless you're very lucky, it's not very easily done, but it's still doable at a bit over that budget with the cheapest, but not awful, components.
If we look at probable cost, thinking that things like saddles and seatposts etc. will most likely be new:
Frame: £15 (old steel conversion etc.)
Wheels: £50 (second hand, could be more like £100, this case including cog + lockring but not always the case)
Tyres: £20
Tubes: £10
Seatpost: £10 (maybe)
Saddle: £20
Stem: £20
Bars: £15
Front brake caliper: £15
Lever: £10
Crankset: £30 (unlikely)
Bottom bracket: £10
(SPD) Pedals: £20
Bar tape: £10
Chain: £8
Total: £263
If we look at probable cost, thinking that things like saddles and seatposts etc. will most likely be new:
Frame: £15 (old steel conversion etc.)
Wheels: £50 (second hand, could be more like £100, this case including cog + lockring but not always the case)
Tyres: £20
Tubes: £10
Seatpost: £10 (maybe)
Saddle: £20
Stem: £20
Bars: £15
Front brake caliper: £15
Lever: £10
Crankset: £30 (unlikely)
Bottom bracket: £10
(SPD) Pedals: £20
Bar tape: £10
Chain: £8
Total: £263