What character is that HTML entity?

JavaScript (ES6), 141 122 134 bytes

a=>a.replace(/&([aeiouyAEIOUY](acute|uml)|[aeiouAEIOU](grave|circ)|[aoAO]tilde);/g,b=>b[1]+{g:"̀",a:"́",c:"̂",t:"̃",u:"̈"}[b[2]])

Followed daavko's example using diacritical marks, and I feel like an idiot for not thinking of using it sooner. Actually gets surprisingly short for JavaScript.

EDIT: Neil caught some bad cases of undefined, which are now fixed.

Retina, 115 bytes

I'm new to code-golf, but I think this might work.
This version was made before the rule, which doesn't allow replacing uppercase html entities (for example Á) was introduced.

i`&([aeiouy])acute;
$1́
i`&([aeiou])grave;
$1̀
i`&([ao])tilde;
$1̃
i`&([aeiou])circ;
$1̂
i`&([aeiouy])uml;
$1̈

Quite simple search and replace. Uses UTF-8.

Uses [letter]\xCC\x[diacritical mark hex code] approach. Diacritical mark is added after every relevant letter.

For some reason, the default Droid Sans Mono font in the interpreter can't render the "circ" and "uml" letters properly. If you change it through developer tools to something like DejaVu Sans, it shows just fine. I think this is a limitation of the font, not the program. But if it's program's fault, I'll try to fix it.

Here is a 129 byte version, which doesn't replace uppercase HTML entites (for example Á)

&([aeiouyAEIOUY])acute;
$1́
&([aeiouAEIOU])grave;
$1̀
&([aoAO])tilde;
$1̃
&([aeiouAEIOU])circ;
$1̂
&([aeiouyAEIOUY])uml;
$1̈

Try it online!
Try it online! 129-byte version

Japt, 81 75 bytes

Ur`&([%vYy](ac©e|uml)|%v(g?ve|circ)|[AaOo]Èìe);`@Yg +'Ì+"?????"g"gutca"bYgJ

The six ?s represent unprintable chars. Test it online!

Note: This outputs the third encoding option; that is, the letter followed by the raw UTF-8 encoding of the corresponding combining diacritical mark.

How it works

Ur"&(    );"       // Replace each ampersand and semicolon that have one of these between them:
([%vYy](acute|uml) //  A vowel or Yy followed by "acute" or "uml",
|%v(grave|circ)    //  or a vowel followed by "grave" or "circ",
|[AaOo]tilde       //  or "a" or "o" followed by "tilde";
@                  // replace each match X and its middle Y with this function:
""g"gutca"bYgJ     //  Take the unprintable at index (index of the second char in Y in "gutca") in this string.
Yg +'Ì+            //  Concatenate the first char in Y and "Ì" to the beginning.
                   // Implicit output

Hexdump of the code:

00000000: 55 72 60 26 28 5b 25 76 59 79 5d 28 61 63 a9 65  Ur`&([%vYy](ac©e
00000010: 7c 75 6d 6c 29 7c 25 76 28 67 9f 76 65 7c 63 69  |uml)|%v(g.ve|ci
00000020: 72 63 29 7c 5b 41 61 4f 6f 5d c8 ec 65 29 3b 60  rc)|[AaOo]Èìe);`
00000030: 40 59 67 20 2b 27 cc 2b 22 80 81 82 83 88 22 67  @Yg +'Ì+"....."g
00000040: 22 67 75 74 63 61 22 62 59 67 4a                 "gutca"bYgJ