As you suspect, that squirrel in your yard is most likely an Eastern Gray squirrel, Sciurus carolinensis. Although they used to be primarily considered an American species, they're naturalized in many countries, and are commonly seen in Europe, especially Great Britain.
Since she nests primarily in large trees, it makes sense that you've been seeing the mother feeding near your biggest trees. As to how far she travels from the nest, I can't say exactly. However, according to the Animal Diversity Organization at the University of Michigan, she has something called a Home Range, (described by some sources as approximately 2.5 acres), which she decreases while she's caring for her babies:
Home ranges are generally larger in the summer. Home range sizes are negatively correlated with squirrel density. Lactating females may decrease their home ranges by as much as 50%. Home ranges are used in the same sequence each day.
Mama squirrel is very resourceful. As you said, the litter may have multiple fathers, none of whom stay once the babies are born. Since her young are in the nest for an average of seven weeks, frequently up to ten, before they're weaned and able to venture out on their own, she's a hard-working single parent! She nurses her babies for long periods of time, keeps them warm, and grooms them. To keep the nest clean she changes their bedding, and even ingests their excrement to prevent poisoning by parasites.
In addition to all that, she has to protect the litter from a variety of predators waiting to raid the nest, some from above, others from below, and she has developed sophisticated tools for doing that.
Two main behaviors are most frequently observed.
When she leaves the nest for any period of time, she covers the babies over with grass, to make them less visible, and therefore less vulnerable.
She frequently builds two nests at the same time, and, if under imminent danger, will move her young.
From New York Wild:
If a mother squirrel perceives a threat to her babies in a nest, she will grab them one by one by the scruff of the neck, and trundle them to a new nest location which may be some distance away. A mother carrying her baby as she scurries along a telephone wire is a memorable sight!
Research studies into squirrel sounds and their meanings are extensive and ongoing, so even though she absolutely has alarm calls, they're not always easy to understand. They depend on many factors, partly including the type and vicinity of the predator. Also, for the first several months after her youngsters are weaned, mom continues to do some surveillance, so her calls might be for protection of those that are independent but still in a maturation process. Some alert calls are used to warn other adult squirrels of the presence of danger, in case they'd prefer to stay away. Those are usually used in her territory but not near the nest.
There appear to be three most frequently used alarm calls.
- Kuks are sharp dog-like barks, usually issued in a series
- Quaas are similar to kuks, but longer
- Moans are lower pitched, and resemble a whistle
Other sounds are called chatter, rattle, muk muk, to name only a few, though they're not specifically indicative of danger.
Tail movements are equally as important when communicating in general, and research shows that certain tail movement/vocalization combinations are likely to indicate danger.
Sometimes she sounds an alarm from the nest. Other times, she purposely separates herself first, so it's hard to know whether or not she'd speak up if you got close. Squirrels are very smart, and also have a keen sense of smell, so if you seem like a curious passerby that doesn't smell like an animal, she's more likely to just sit on or at the nest. If you reach out to touch it, that would be a different story, but since that's not your intention, time spent watering or walking around your trees is probably not enough to cause her to make much noise.
Links to further information, many of which have recordings of squirrel sounds:
Cornell Lab of Ornithology Macaulay Library
Squirrel Alarm Calls are Surprisingly Complex
How to Talk Squirrel
Soundboard Squirrel Sounds (Only some of these are free, but those are worth a listen.)
Discover British Wildlife Magazine
New York Wild Organization (This site also has live webcams.)
Smithsonian Museum of Natural History
If you're really interested and want some "light reading" check out this 194 page dissertation of squirrel calls, published in 2012 by the University of Miami, Florida. I haven't read it!
There are a number of live webcams online. Look around and you'll see some fascinating things.