Well, it looks like at the least you could benefit from combining some lines. There's a few places that you define a variable, and then immediately redefine it by filtering itself. Such as:

$servs = $all_computers | Select-Object Name,DistinguishedName
$servs = $servs -Match "Servers" 

Can be reduced to:

$servs = $all_computers | Select-Object Name,DistinguishedName | Where {$_ -match "Servers"}

Now do that same thing for $admin_exist_cnt and $upep_admin_exist_cnt

$admin_exist_cnt = Compare-Object -DifferenceObject $group_name -ReferenceObject $com_name_admin -ExcludeDifferent -IncludeEqual -Property name | select-object @{name="name";expression={$($_.name).toString().TrimEnd("A","d","m","i","n","s")}}

and

$upep_admin_exist_cnt = Compare-Object -DifferenceObject $group_name -ReferenceObject $com_name_upep -ExcludeDifferent -IncludeEqual -Property name | Select-Object @{name="name";expression={$($_.Name).ToString().TrimEnd("U","P","E","P","A","d","m","i","n","s")}}

Then near the end you run through all the computers looking for "both" admintypes, and remove those from the $all_computers variable, then run through the whole thing again, and do that like 4 times. No, don't do that, use the Switch command instead. It would look like this:

Switch($all_computers){
    #Finding all of the accounts where both types of admins exist and removing them from the master list
    {$both_exist_cnt.name -contains $_.name} {$_ | Add-Member "admintype"  "both";Continue}
    #Finding all of the accounts with only exception admins and removing them from the master list
    {$admin_exist_cnt.name -contains $_.name} {$_ | Add-Member "admintype" "old";Continue}
    #Finding all of the accounts with only upep admins and removing them from the master list
    {$upep_admin_exist_cnt.name -contains $_.name} {$_ | Add-Member "admintype" "UPEP";Continue}
    #These accounts have no admin
    default {$_ | Add-Member "admintype" "No"}
}

Then you can reduce the output section to just:

#output
$all_computers | Export-Csv test.csv

Edit: Ok, went back over things and saw that you're redefining things a whole lot. Instead I propose just running it through a Switch once, then counting up your results afterwards. This will reduce memory consumption considerably, which may not matter on small test runs but should make a considerable difference when running against large quantities. Try this revised script, and let me know if you have any specific questions:

#TODO Fix the server thing and put it in a for loop
$all_computers = Get-ADComputer -Server NW -filter * -Properties enabled | select name,enabled,distinguishedname
#Gets all of the objects that are in a group and pulls their names this is to get the admin flag that is appended to names in this group
$group_name = Get-ADObject -Server NW -Filter{objectCategory -eq "group"} | select name

#Counts the devices pulled from the computer query
#TODO Replace "test.txt" with a descriptive file name
$DevNum = $all_computers.count
echo "Number of devices: " $DevNum > test.txt

#Remove servers from the list
$all_computers = $all_computers | ?{ $_ -NotMatch "Servers" }

#Counts the number of servers we removed
$NumSkipped = $DevNum - $all_computers.count

Switch($all_computers){
    #Finding all of the accounts where both types of admins exist and removing them from the master list
    {$group_name.name -contains $($_.name + "Admins") -and $group_name.name -contains $($_.name + "UPEPAdmins")} {$_ | Add-Member "admintype"  "both";Continue}
    #Finding all of the accounts with only exception admins and removing them from the master list
    {$group_name.name -contains $($_.name + "Admins")} {$_ | Add-Member "admintype" "old";Continue}
    #Finding all of the accounts with only upep admins and removing them from the master list
    {$group_name.name -contains $($_.name + "UPEPAdmins")} {$_ | Add-Member "admintype" "UPEP";Continue}
    #These accounts have no admin
    default {$_ | Add-Member "admintype" "No"}
}

#TODO Replace "test.txt" with a descriptive file name
echo "Number of servers skipped: " $NumSkipped >> test.txt

#Calculating the number of workstations 
echo "Number of workstations: " $all_computers.count >> test.txt

#TODO Replace "test.txt" with a descriptive file name
echo "Number of Exception admins found: " $($all_computers|?{$_.admintype -match "old|both"}).count >> test.txt

#TODO Replace "test.txt" with a descriptive file name
echo "Number of UPEP admins found: " $($all_computers|?{$_.admintype -match "upep|both"}).count >> test.txt

#Find both exception and upep admin names
if($($all_computers|?{$_.admintype -ne "No"}))
{
        echo "Number of both UPEP and Exception Admins: 0" >> test.txt
}else
{
        echo "Number of both UPEP and Exception Admins: " $($all_computers|?{$_.admintype -match "both"}).count >> test.txt
}

#output
$all_computers | Export-Csv test.csv