Google Sheets, 26 bytes

Thanks to my colleague Richard for spotting an error.

Takes input in cell A1.

=TEXT(1+A1,"d""th"" mmmm")

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Google Sheets represents dates with 1 being 31st December 1899, so this adds one to the day number and formats appropriately.

The formula does not work in Microsoft Excel because it counter-factually calculates 1900 as a leap year.

Hack, 115 59 39 bytes

$x==>date("jS F",mktime(0,0,0,1,$x));

Since @gwaugh got to the same solution as mine while I was golfing, I'm posting this in Hack instead :).

Python 3.8 (pre-release), 112 bytes

lambda x:str(d:=(t:=gmtime(x*86399)).tm_mday)+'tsnrhtdd'[d%5*(d%30%20<4)::4]+strftime(' %B',t)

from time import*

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Weirdly enough, I don't have to parenthesize d:=(t:=gmtime(~-x*86400), probably because the interpreter only checks if there are () characters around the assignment expression and not that the expression itself is parenthesized.

-2 thanks to gwaugh.

-5 thanks to xnor.

PHP, 38 40 30 28 bytes

<?=date("jS F",86399*$argn);

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Run with php -nF input is from STDIN. Example (above script named y.php):

$ echo 1|php -nF y.php

1st January

$ echo 2| php -nF y.php

2nd January

$ echo 3| php -nF y.php

3rd January

$ echo 11|php -nF y.php

11th January

$ echo 21|php -nF y.php

21st January

$ echo 60|php -nF y.php

1st March

$ echo 365|php -nF y.php

31st December

Explanation

Construct an epoch timestamp for the desired day in 1970 (conveniently not a leap year) by multiplying the day number * number of seconds per day (86400). However, this would yield one day higher so instead multiply by number of seconds in a day - 1 (86399) which for the range of input numbers (1≤n≤365) will result with the timestamp of the end of each correct day. Then just use PHP's built-in date formatting for output.

JavaScript (ES6), 117 bytes

d=>(n=(d=new Date(1,0,d)).getDate())+([,'st','nd','rd'][n%30%20]||'th')+' '+d.toLocaleDateString('en',{month:'long'})

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Commented

d =>                     // d = input day

  ( n =                  //

    ( d =                // convert d to

      new Date(1, 0, d)  //   a Date object for the non leap year 1901

    ).getDate()          // save the corresponding day of month into n

  ) + (                  //

    [, 'st', 'nd', 'rd'] // ordinal suffixes

    [n % 30 % 20]        // map { 1, 2, 3, 21, 22, 23, 31 } to { 'st', 'nd', 'rd' }

    || 'th'              // or use 'th' for everything else

  ) + ' ' +              // append a space

  d.toLocaleDateString(  // convert d to ...

    'en',                // ... the English ...

    { month: 'long' }    // ... month name

  )                      //

Without date built-ins, 188 bytes

f=(d,m=0)=>d>(k=31-(1115212>>m*2&3))?f(d-k,m+1):d+([,'st','nd','rd'][d%30%20]||'th')+' '+`JanuaryFebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember`.match(/.[a-z]*/g)[m]

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C# (Visual C# Interactive Compiler), 141 139 133 124 122 bytes

a=>{var d=s.AddDays(a-1);int x=d.Day,m=x%30%20;return x+"thstndrd".Substring(m<4?m*2:0,2)+d.ToString(" MMMM");};DateTime s

Thanks to Arnauld for faster method of removing 11,12,13th saving 4 bytes

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Perl 6, 228 185 bytes

{$_>365&¨~Date.new(Date.new(Date.today.year,1,1).later(days=>$_),:formatter{.day~[0,"st","nd","rd",|("th"xx*)][.day%30%20]~' '~ [Date::Names.new.mon(.month)]})}(243)

Error checking costs 12 bytes.
AFAIK Perl 6 doesn't have month names built-in :. Not sure I'm allowed -MDate::Names or not.
.later is currently bugged on rakudo and drops the :formatter, so I need to re-create a Date on top with the correct formatter.

C# (Visual C# Interactive Compiler), 115 bytes

n=>{var g=p.AddDays(n-1);int f=g.Day,m=f%20%30<4?f%10:0;return f+$"{"tsnr"[m]+""+"htdd"[m]} {g:MMMM}";};DateTime p;

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R, 158 134 bytes

-24 bytes @Nick Kennedy for golfing the 'st', 'nd', 'rd', & 'th'. Thanks!

f=format;paste0(a<-as.double(f(d<-as.Date(scan(,''),'%j'),'%e')),`if`((a-1)%%10>2|a%/%10==1,'th',c("st","nd","rd")[a%%10]),f(d,' %B'))

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Perl 6, 166 163 bytes

{.day~(<th st nd rd>[.day%30%20]||'th')~' '~ <January February March April May June July August September October November December>[.month-1]}o*+Date.new(1,1,1)-1

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When you hardcode the month names, but still beat the previous Perl 6 answer that didn't >.<

Jelly, 115 114 bytes

’%⁵<3a:⁵n1Ɗ×%⁵$Ḥ+ؽị“thstndrd”ṭ

“5<Ḟ’b4+28ÄŻ_@µ>0T,>0$ƇZṪµ1ịị“1;8-UÞẆĊƬṁɱkzIṇkµ⁴ JḷJdɱ<©Q£Ƈ.dẊṘh!İḞẠƒĊ¶÷»Ỵ¤,2ịÇƊṚK

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Long by Jelly standards, but done from first principles.