As your kitchen style is quite traditional with wooden counter-top and Belfast sink, I would go with something simple like subway tiles which don't date. Alternatively you could use a small stainless steel upstand running along the back of the counter and a bit further up behind the hob/. This would not draw too much attention to itself and would tie in with the extractor fan hood.

If you fancy something more exciting you could go with some colourful tiles. A friend of mine has a kitchen almost identical to yours and her backsplash is colourful mismatched Mexican tiles. It's a strong statement but looks fabulous and very cheerful.

As for height, you could go full height if using something neutral like white subway (it can be a little tricky I believe though to tile behind an extractor hood/ vent that has already been fitted). In an 11ft ceiling tiles all the way up might look a little overwhelming (and be quite expensive). You could just tile as far as the lowest shelf but behind the cooker I'd take the tiles up to the extractor hood or it might look funny. Normally, if there are wall cabinets they're around 50cm from the counter-top and people tile this area, going further up behind the hob.

Your kitchen is very nice btw!

Don't be swerved away from hand painted tiles! The best way to add colour and originality (I make hand painted tiles so I may be biased).
I would take the tiles to just under the bottom shelf, and up to the hod.
We have had tiles behind our cooker for 21 years, and we don't find the grout staying clean a problem.

By the way - I think subway tiles will look dated in a few years, as anything bang on trend tends to do.

mosaic every time - this is one that i did in my own kitchen behind our cooker (totally flat so easy to keep clean - and we run a B&B too). the rest of the kitchen can be complementary plain tiles... personally a bit of colour is always good

Hand painted tiles as a feature over your hob up to the hood, then 1 tile high across the work top as a patterned upstand. In my kitchen I painted the walls in wipeable paint and just used a wall sticker behind my hob - it says 'Eat Drink and Be Merry'! There are some great ones on Amazon.

I would say either subway tiles, handmade tiles, or even marble depending on your style. I would only tile up about 20-30cm high (or halfway up to the bottom shelf), and the up to the extractor behind the cooker. I think if you tile all the way up to the bottom of the shelf it could get messy with the brackets. Proportionally I think halfway up would look right.

Thank you for all the helpful comments!