The legal implications are negligible. A ceremonial declaration that English is official has no practical effect; other legislation is required to have an effect.
Congress could pass a law declaring that English is the official language of the US, but they would also have to pass additional legislation for that to have any effect. For example, to prevent a bill from being introduced in Congress in Turkish, they would have to pass a law requiring all bills to be written in English (or not in Turkish). It is nevertheless a fact that all bills are written in English, without such a law and without an official declaration that English if the official language.
Congress could pass a law withholding funding from any educational institute which teaches in a language other than English (teaches any students), but such a law would also require repeal of various anti-discrimination laws whereby children have a right to native-language education (to a point); patients have a right to a translator; witnesses and the accused have a right to a translator in official hearings. But: all of this could be done without passing a law making English the official language.
The US could also adopt a constitutional amendment, similar to section 3 of the Spanish constitution ("Castilian is the official Spanish language of the State. All Spaniards have the duty to know it and the right to use it"), which means that a Catalan or Basque-speaking Spaniard has no right to a translator in a trial). Even if there were a constitutional amendment saying that English is the official language, we we still have to await myriad SCOTUS interpretations of that clause, to determine whether that prohibited immigration agents from using a language other than English when addressing a customer.
Individual states can have different policies (summary here), where English and Hawaiian are official in Hawaii, English and a couple dozen native languages are official in Alaska (Inupiaq, Siberian Yupik, Central Alaskan Yup'ik, Alutiiq, Unangax, Dena'ina, Deg Xinag, Holikachuk, Koyukon, Upper Kuskokwim, Gwich'in, Tanana, Upper Tanana, Tanacross, Hän, Ahtna, Eyak, Tlingit, Haida, and Tsimshian). I understand that laws are published in English and Spanish in New Mexico. Such laws and practices would still be possible if English were declared the official language of the US, by act of Congress. Tennessee has a fairly strong official-English law:
English is hereby established as the official and legal language of
Tennessee. All communications and publications, including ballots,
produced by governmental entities in Tennessee shall be conducted in
English unless the nature of the course would require otherwise.
Because this law not only makes English the official language but also mandates that English be used for government publications "unless otherwise required", that does mean that you can't take your driving test or get a ballot in Farsi or Spanish (at least, by the letter of the law), as you can in some stated. This page summarizes the state-by-state situation.
The California Constitution says that English is the official language of the state. But the core mandate of the clause is fairly toothless:
The Legislature shall enforce this section by appropriate legislation.
The Legislature and officials of the State of California shall take
all steps necessary to insure that the role of English as the common
language of the State of California is preserved and enhanced. The
Legislature shall make no law which diminishes or ignores the role of
English as the common language of the State of California.
Nevertheless, they produce voters pamphlets in 7 languages and driver's knowledge tests in Amharic, Arabic, Armenian, Cambodian, Chinese, Croatian, French, German, Greek, Hebrew, Hindi, Hmong, Hungarian, Indonesian, Italian, Japanese, Korean, Laotian, Persian/Farsi, Polish, Portuguese, Punjabi, Romanian, Russian, Samoan, Spanish, Tagalog/Filipino, Thai, Tongan, Turkish, and Vietnamese.
The political consequences would not be at all trivial.