Does any English dialect use any non-English foreign letters in their alphabet?

The usual pattern is that words that are still perceived as foreign retain their original spelling, but if they become a part of the English lexicon, the spelling is modified to a more English-seeming orthography. For instance, Conan Doyle used the spelling cañon at around the beginning of the twentieth century for a word we now spell as canyon. Where the characters are truly uncommon, English has always tended to transliterate into existing symbols.

We have and use diacriticals in English, although they seem to have fallen out of use since the introduction of word processing. One still often sees a diaeresis used in naïve or noël, but one rarely sees coöperate anymore. We also use ligature graphemes like ash (Æ or æ), but the trend has been toward either dropping one of the two ligated letters or setting the letters separately.

There are some loanwords from French, some of them preserving original spelling. For example "façade" (although in English spelling "facade" is also accepted).

Another thing that comes into mind is load from German, prefix "über-", although it's not really part of official language.

In science -- 1Å (ångström) == 0.1nm. Not sure if you'd count that as part of language...

Not sure where you'd put Æ, after all it's not foreign, it's Old English.

Does any of them include or plan to include foreign letters?

It is possible that English alphabet will include letters that are not included in the today alphabet, in the same way English passed from the alphabet used in Old English to the nowadays alphabet. I cannot say it is planned, as English doesn't have an academy (or any authoritative agency, see Regulatory bodies and authoritative dictionaries for English) that plans such things, or proposes those things.

Well, there are no letters in the Latin alphabet which are missing from English, so there is nothing more to add. On the contrary, English used to have more letters in the past, such as eth and thorn. They are not likely to come back, I presume.

As for letters with diacritics, there are several words, primarily (or always?) borrowed from French, that are often written in English with diacritics, such as naïve, naïveté, fiancé.