Make a numeric list converter

JavaScript (ES6), 75 82

As an anonymous function

Edit: 2 byte saved thx @user81655 (and 5 more just reviewing it)

(l,[a,b])=>a.trim()+l.match(/[-\d.]+/g).join(b)+']})> '['[{(< '.indexOf(a)]

Test snippet

F=(l,[a,b])=>a.trim()+l.match(/[-\d.]+/g).join(b)+']})> '['[{(< '.indexOf(a)]

// Test
console.log=x=>O.innerHTML+=x+'\n'
// default test suite
t=[['1 2 3 4','[,'],['<1;  2;  3>',' ;'],['{-1.3, 3.4, 4, 5.55555555}','[,']]
t.forEach(t=>console.log(t[0]+' *'+t[1]+'* '+F(t[0],t[1])))
function test() { console.log(P1.value+' *'+P2.value+'* '+F(P1.value,P2.value)) }

#P1 { width: 10em }
#P2 { width: 2em }

P1<input id=P1>
P2<input id=P2>
<button onclick="test()">-></button>
<pre id=O></pre>

CJam, 35 34 bytes

l(S-l"{[<(,}]>);":BSerS%@*1$B5/~er

Test it here.

Expects the format on the first line and the list on the second.

Explanation

l   e# Read the format line.
(   e# Pull off the first character, which is the opening bracket.
S-  e# Set complement with a space, which leaves brackets unchanged and turns a space
    e# into an empty string.
l   e# Read the list.
"{[<(,}]>);":B
    e# Push this string which contains all the characters in the list we want to ignore.
Ser e# Replace each occurrence of one of them with a space.
S%  e# Split the string around runs of spaces, to get the numbers.
@   e# Pull up the the delimiter string.
*   e# Join the numbers in the list with that character.
1$  e# Copy the opening bracket (which may be an empty string).
B5/ e# Push B again and split it into chunks of 5: ["{[<(," "}]>);"]
~   e# Unwrap the array to leave both chunks on the stack.
er  e# Use them for transliteration, to turn the opening bracket into a closing one.

Pyth, 33 bytes

rjjezrXwJ"<>[]  {}(),;"d7@c6JChz6

Try it online: Demonstration or Test Suite

Explanation:

J"<>[]  {}(),;"  assign this string to J

rjjezrXwJd7@c6JChz6   implicit: z = first input string, e.g. "[;"
       w              read another string from input (the list of numbers)
      X Jd            replace every char of ^ that appears in J with a space
     r    7           parse ^ (the string of numbers and spaces) into a list
  jez                 put z[1] (the separator symbol) between the numbers
            c6J       split J into 6 pieces ["<>", "[]", "  ", "{}", "()", ",;"]
               Chz    ASCII-value of z[0] (opening bracket symbol)
           @          take the correspondent (mod 6) brackets from the list
 j                    and put the numbers between these brackets
r                 7   remove leading and trailing spaces