The prospect of leftover anything (including pudding*) for breakfast brings me much joy, & New York has proved particularly glorious in enabling my habit. In England doggy bags are the exception not the rule and restaurants, particularly outside of London, do not appreciate being asked to wrap up your leftovers - I’ve found waiters will often say that the dread Health & Safety regulations won't allow it and, if pushed, will present you with a mound of food wrapped in leaky aluminium foil to take home - whereas there doesn’t seem to be a restaurant in Manhattan that would object.
In New York I’ve doggy bagged everywhere from Le Bernadin to my local diner, and rather like the get two meals for the price of one that results from having the balls to ask. (Altho apparently if you ask for a doggy bag on dates 1-3 with a new man, he will take that as sign that you are not interested. Or just more interested in the free food.)
The difference stems partly from English restaurant portions being human sized (as opposed to dinosaur sized), so there is usually little need to halve your food and save the rest for later, but I suspect the main reason is that takeout food is generally the sole province of ethnic places and fast food outlets, so restaurants just aren't used to packing up food to go.
Using a restaurant is an occasion for most people, rather than an everyday thing as it is for many Americans: it’s just way, way more expensive in the UK to eat out. We don’t have the equivalent of family friendly diners where anyone from a truck driver to an office worker can get a good, cheap hot meal. This emphasis on restaurants being for occasions means that asking for your leftovers to be wrapped can be considered distinctly down market, and results in having some over-bearing waiter looking down their nose at you.
*Have just discovered – after two years of NY living - that pudding over here is the sweet,creamy chilled stuff that comes in pots with a foil lid in the grocery store, rather than being a catch-all name for the sweet course at the end of the meal.