My baby is tall, 16 months old, her height is over 85 cm, weighs 13 kg, and our current Giggle 2 Cossatto pushchair starts to be too low, her head almost reaches the backrest's metal frame. I am looking for a pushchair with a backrest that's higher than that, one that she's going to be comfortable having a nap in and preferably under £150. Is there a brand/model you could recommend?
For this size and age, I would certainly recommend something smaller and world-facing only. That will provide much more space for your girl and will be lighter as well as more affordable than any reversible stroller.
From the truly small ones, do consider a Chicco Goody (Chicco Goody full stroller review here, if interested), or a Mee-go Trio Plus - there will be enough space, but still, these, you need to take as small buggies, so don't expect them to perform fantastic on terrain nor to have super long seat surface. They will last until 3yo, however, if necessary.