So the suggestions seem to be the NEX system or the micro 4/3 system.
The fixed lens cams I looked all missed the longer focal length requirement. (Or not?)
So just to get a rough idea of size and weight here are the empty body weights and lens weight for lens(systems)giving 28-250mm (35mm equiv) focal lenths.
The Sony Bodies are thinner than the Olympus ones I think, but their lenses longer.
pana 14-140mm OIS, 460g
pana 14-42mm OIS,95g, 27mm long!
Minimum weight: 722g
Olymp 12-50mm 211g, 83mm long, dust and spray resistant
Olymp 14-42mm II R, 113g, 50mm long
Olymp 14-150mm, 280g, 83mm long
Olymp 40-150mm, 190g, 83mm long
OM-D OIS 365g
Minimum weight: 645g
Sony 18-200mm OIS, 524g, 102 mm long!
Tamron 18-200 OIS, 460g
Sony 18-55mm OIS, 194g, 60mm long
Sony 55-210mm OIS, 345g, 108mm long!
Sony 16mm,70g, 23mm long
Minimum weight: 751g
For comparison lightweight APSC SLR:
Sony 18-250mm 440g , this is a larger zoom range(15x vs 10x), a 18-200 is 40g less
Sony a500 DSLR 596g
This is about 300-400g more.
It would seem that the biggest gains for the mirrorless systems are NEX with a pancake lens, where the flat package might fit in a pocket, and m4/3 for longer telephoto lenses where the smaller glass can be a fair bit lighter.