How can I best approximate the usual taste and texture without any tomato ingredients?
The tomato in shepards pie provides acid and is the main coloring agent of the meat portion of the dish. Any other acid will work in place of it, although which you use will entirely depend on how much you want the flavor profile to stay the same. Adding lemon juice will give your pie a Mediterranean taste, adding others will result in other flavors coming to mind. You could just leave the tomatoes out and use a roux instead of the flour, that should give you about the same mouth feel just tasting onions and lamb.
You could prop ably best approximate the flavour of the tomatoe by using some sort of roast pepper paste or roast peppers - although how well this will go in a shepherds pie I'm not so sure. If you just want the color you could use some paprika but again in a shepherds pie I'm not so sure!
Hope this helps!