Where to pee in London?

I have not been to London with kids but according to my experience there are plenty of public toilets available. Most parks have facilities that include a café and toilets.

In the city there are toilets on most big squares. There's always a McDonald's restaurant nearby where you can go.

One advise is not secret but always given: go to a museum. In many entrance is free and so is the use of the toilets.

Is it acceptable to ask businesses for kids to use their toilets, even if we are not customers?

It is if you ask first.

It's also not that difficult to become a customer. Go to a café, order a cup of coffee or an ice-cream. Nobody expects you to order a three-course meal to let you use the bathroom.

If you absolutely don't want to buy anything, you can just keep asking "can my kid use your bathroom" in a few places till you find one that lets you in for free. From my experience, you shouldn't be searching for long.

Is it acceptable for small children to pee (only pee) on trees and such in parks?

No.

Pro tips:

1. Escorting your kids to the toilet before leaving a place with one (even if your kids say they don't need to go) helps to avoid looking for one in the next couple of hours.

2. Ultimately, if your kids can't wait even 5 minutes before they have to go, consider getting some diapers. Walking through a city while constantly paying attention to be within 5 minutes from a toilet facility is not exactly a pleasant tourist experience.

They need to go now.

As the parent of an 8-year-old, the most successful approach about age 4-5 was to insist he went to the toilet at intervals which fitted what we were doing, even if he claimed to not need the toilet. If he genuinely didn't, that was fine - just give him 5 minutes and then move on. Often he did though. Now it's a case of him saying "I don't really, but I'll try", because he's learnt it's a good idea.

Re bushes in parks, it's never good, but whether it's even a bit OK depends on the park's wildness, how old the child is, and how discreetly it can be done. If it's a deer park, chances are no-one would notice an adult going behind a tree. In the middle of Kew Gardens is not the same thing! And whilst people appreciate a 3-year-old will have emergencies, a 6-year-old will not get the same leeway. The key part though is that you must be clearly attempting to make it as discreet as possible. If you don't seem to care about your kids peeing in public, however young they are, you will find yourself in trouble. Not just disapproval trouble, but police trouble.

To add to all of the above, there are pubs everywhere - and there are always toilets in pubs. In my years of living and working in London, I have many times used a toilet in a pub without being their customer. Now, these toilets may not be the cleanest around, but "when you gotta go, you gotta go".

The Great British Public Toilet Map might be able to help find toilets in the UK: https://www.toiletmap.org.uk/

You might be able to find similar such maps in other places you travel. The Australian Government, for example, maintains the National Public Toilet Map https://toiletmap.gov.au/

These are some of the common places that you can find public restrooms.

Restaurant/Cafes/bars - Starbucks, McDonald, KFC, etc.
Public parks
Hotels/Hotel lobbies - In some hotels, you might need a hotel card to get in.
Public buildings & Transport hubs - Libraries, Hospitals, Train stations
Shopping Malls & Department stores

You can also do a simple online search or you can use Google map to find public restrooms. If you can't, you can use these apps to find places that have public restrooms such as a restaurant (Starbucks, McDonald, etc) or you can use one of the dedicated apps as suggested in other answers.

London is full of busy pubs and McDonalds - you can use either without worry. Toilets in the terminals are also free.

I've used this advice for 15 years to function effectively in cities.

Plan your eating

An essential strategy in any city is to plan your eating so that at times you will need the bathroom, you will be having a meal or snack. That makes you a paying customer, and you are definitely entitled to use any bathroom they have for customers.

Partner to this strategy is do not gorge yourself. Eat lightly so you have room for another snack at another establishment should the "need" arise. To be clear, the "snack"'s solitary purpose is to provide a statutory transaction for use of the bathroom; it can be one biscuit that you share or simply throw away. Since you have no issue with paying for bathroom access, this shouldn't be a problem.

Plan your drinking

This should go without saying, but err on the side of drinking less when you're out and about. You don't need to pee what you didn't drink. The wrong thing is to make full use of free refills, or get the large because it's twice the drink for 10% more, for instance, so you have to re-train those thrifty habits.

And sync your drinking up with your kids' so you're drinking about the same in proportion. That way you'll know how they feel.

Since dehydration is a concern -- actual, medical dehydration can be trivially tested in 2 seconds. Ask your doctor. So if you are worried about this, learn the test and do it as needed. If you are adequately hydrating your children normally, the limiting of fluids a few hours before and during the city excursion will be inconsequential, unless they get overheated, which you'll know because you'll also be overheated. If in doubt, test for it.

Hotels

Most city hotels have conference rooms and they are usually renting different rooms to different companies at once. So the areas are public access. Just walk in like you know where you're going and seek out that area. Bathrooms are nearby.

Train your kids to go when able

Break them out of the habit of only thinking about bathrooms when nature calls. Teach them to go when the opportunity arises. This is a valuable life skill that you'll be giving them.