All of my DCs went through the US early elementary system and all emerged very strong students, great readers and also good at maths. Once your DD gets to kindergarten the class will start on phonics and it will 'take' fast unless there's a problem.
The fretting of British parents about early reading, etc., is not necessary. The UK is the only country where such an early start to formal teaching is mandated, and results at the other end do not bear witness to anything but the futility of the approach. Some students do very well (see BBC report stating that 5 schools sent more pupils to Oxbridge than 2000 other schools combined, over the last three years) but there is huge group of underachievers at the lower end.
Students exposed to phonics early (as in the UK) may or may not learn, and may or may not be able at that early age to deal with the sense of failure to do something they would love to do in order to please mommy or teacher, but which they find incomprehensible, at 4 or so, whereas they may be ready at a later age to learn much faster. At a later age they are more emotionally resilient, and can deal with frustration a bit better. They have usually developed more of a sense of themselves as competent little people and are not as depressed about finding one aspect of their lives challenging as they would be at age 4. They are more likely to have already formed a positive self image and to believe in their competence from 5 to 7. At age 4 it is quite risky to have a child see himself as a failure.
Most preschools in the US will encourage parents to read, read, read to their children, to encourage pre reading activities like puzzles, simple board games, listening activities, singing, dance and physical co-ordination activities, music, scribbling, colouring, painting, playing with sand and water, blocks -- simple play that encourages active participation, motor skills, planning, co-operation and self direction in other words.
A lot of the preschool regimen in the US is devoted to training the children in good classroom habits -- listening, politeness, tidiness, self care, co-operation and sharing with other children, problem solving, getting used to raising a hand to speak. My own DCs always had 'show and tell' in pk4 and Kdg, which was a very useful oral learning activity. The little talk had to be planned and practiced and then each student had to speak and answer questions about their object.
Self-care was also very much encouraged -- parents were not allowed into the school in the morning or afternoon. We said our goodbyes at the door. The DCs were responsible for taking care of all their own coats, mittens, hats, snowboots (cold winter climate) both taking off and putting back on to go home, with all zipping, buttoning, tying that that involved. The benefits of all that competence and regimentation are reaped in First Grade, where work tends to be very productive.
Trust your DCs' teachers. There is a purpose to it all and your DCs are not spinning their wheels.
Some of my DCs learned to read all by themselves without any phonics instruction, just by being read to as far as I could see, plus watching Sesame Street, etc., as early as age 3.5 and 4. They benefited from the later exposure to the systematic phonics and reading/writing/spelling programme when they entered Kdg and first grade and on through the elementary grades. Those of the DCs who learned at age 5ish progressed at the same rate. They were all exposed to the same reading at home, but learned at different ages.