Ruby, 139 bytes

Felt nostalgic and decided I wanted to golf down my first ever answer on this site with everything I've learned since then. I wasn't even using stabby lambdas or basic idioms like map in place of for loops for that answer!

->x,y{s=' ';puts s+?_*x[0]+s
y.zip(x,x[1..]){|i,n,m|puts"|#{s*n}|
"*~-i+?|+s*[n,m||=0].min+(m<1??_:m>n ??|:s)+?_*~-(m-n).abs+(m<n ??|:'')}}

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Ruby 191

First time golfing, also it's my first day with Ruby, so it's probably not the most elegant thing in the world, but it'll do?

def f(x)
a=x[0]+[0]
puts" #{'_'*a[0]} "
for i in 0..x[1].length-1
n,m=a[i,2].sort
puts"|#{' '*a[i]}|\n"*(x[1][i]-1)+'|'+' '*n+(a[i+1]<1?'_':m>a[i]?'|':' ')+'_'*(m-n-1)+(n<a[i]?'|':'')
end
end

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Retina, 169 150 113 bytes

Byte count assumes ISO 8859-1 encoding.

\d+
$*
{+r`1(1*¶[^|]*(1+))
$1¶|$2|
}` (¶.*) 1+
$1
1
_
(\|_+)\|(?=¶\1_(_+))
$1|$2
T`w` `(\|_+)_?(?=_*\|.*¶\1)
^¶
 

The code contains a trailing space on a trailing newline.

Input format:

Height (separated by spaces)
Width (also separated by spaces)

For example:

1 2 3 1 2 1 1 1 2 1
2 14 6 8 4 18 2 10 4 2

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Python 3, 230 223 222 217 bytes

def f(a):b=a[0]+[0];m=' _';print('\n'.join([' '+'_'*b[0]+' ']+['|'+' '*min(b[l],b[l+1])+m[b[l+1]<1]*(b[l]>b[l+1])+m[k]*(b[l]-b[l+1]-1)+'|'+m[k]*(b[l+1]-b[l]-1)for l,i in enumerate(zip(*a))for k in[0]*(i[1]-1)+[1]]))

Thanks to @StewieGriffin @KevinLau for their help

Results

>>> f([[2, 14, 6, 8, 4, 18, 2, 10, 4, 2],[1, 2, 3, 1, 2, 1, 1, 1, 2, 1]])
 __ 
|  |___________
|              |
|       _______|
|      | 
|      | 
|      |_
|     ___|
|    |             
|    |_____________
|   _______________|
|  |_______
|     _____|
|    |
|   _|
|__|

JavaScript (ES6) 174

The only critical part is the horizontal row joining 2 parts of different widths, with the vertical bar on right side that can be in the middle or at the right end.

(p,h,q=-1,R=(n,s=' ')=>s.repeat(n))=>[...p,0].map((x,i)=>(x>q?p=x:(p=q,q=x),(~q?`
|`+R(q+(x<p)-!x)+R(x>q,'|'):' ')+R(p+~q+!x,'_')+R(x<p,'|')+R(h[i]-1,`
|${R(q=x)}|`))).join``

TEST

f=(p,h,q=-1,R=(n,s=' ')=>s.repeat(n))=>[...p,0].map((x,i)=>(x>q?p=x:(p=q,q=x),(~q?`
|`+R(q+(x<p)-!x)+R(x>q,'|'):' ')+R(p+~q+!x,'_')+R(x<p,'|')+R(h[i]-1,`
|${R(q=x)}|`))).join``

// Less golfed

F=(p,h, q=-1, 
   R=(n,s=' ')=>s.repeat(n)
  )=>
  [...p, 0].map((x,i)=> (
    x>q? p=x : (p=q,q=x),
    (q>=0?`\n|`+R(q+(x<p)-!x)+R(x>q,'|'):' ')
    + R(p+~q+!x,'_') + R(x<p,'|')
    + R(h[i]-1,`\n|${R(q=x)}|`)
  )).join``

console.log=x=>O.textContent+=x+'\n'

;[  
  [[2],[1]],
  [[4],[2]],
  [[2, 6, 2, 4],[2, 2, 1, 3]],
  [[2, 14, 6, 8, 4, 18, 2, 10, 4, 2],[1, 2, 3, 1, 2, 1, 1, 1, 2, 1]]
].forEach(t=>{
  var w=t[0],h=t[1]
  console.log('['+w+'] ['+h+']\n'+f(w,h)+'\n')
})

<pre id=O></pre>