How to download or export the budget details from azure portal or using PowerShell which are created in root scope?

First of all, if you want to see the root and the resources in it, you need to enable the "Access management for Azure resources"

1. Open the Azure portal
2. Search "Tenant properties"
3. Turn on the "Access management for Azure resources"
4. Relogin

Reference and help: Elevate access to manage all Azure subscriptions and management groups

"However, when management groups are enabled for the organization, all
subscription costs are rolled-up to the billing account and to the
root management group because they’re both constrained to a single
directory."

"Management groups – Hierarchical containers, used to organize Azure
subscriptions. A management group tree can support up to six levels of
depth. The limit doesn’t include the Root level or the subscription
level."

Reference: Understand and work with scopes

So I think that what you want in this context is not achievable.

But! You can use the Get-AzManagementGroup command to query all subscriptions within the root.
You can then iterate through all the subscriptions (Set-AzContext) and then query the budgets (Get-AzConsumptionBudget).

You can of course store these in an object on the fly and save them to a csv (or whatever) file at the end.

Something like this:

#Make sure that you have Az module installed
#If you don't have enough permissions, you can just delete or comment out the lines with "#permission" at the end.

#install-module az
    #Or if you don't need all module just the necessary ones run the following commands
#install-module az.billing
#install-module az.accounts
#install-module az.resources
#install-module az.accounts

Connect-AzAccount

$allTenant = Get-AzManagementGroup #permission

$arrayToFile_AllDetails = @()
$arrayToFile_MinimalDetails = @()

foreach($actualTenant in $allTenant){ #permission
    $allSubInTheTenant = Get-AzSubscription -TenantId $actualTenant.TenantId #permission, just delete "-TenantId $actualTenant.TenantId"

    foreach($actualSub in $allSubInTheTenant){
        Set-AzContext -Subscription $actualSub.Id #| out-null #you can add the "| out-null" if you don't want to see when switch between subs
        $budget = Get-AzConsumptionBudget
    
        if($budget -ne $null){
            $minimalObject = [PSCustomObject]@{
                BudgetName = $budget.Name
                Amount = $budget.Amount
                Category = $budget.Category
                TimeGrain = $budget.TimeGrain
            }

            $arrayToFile_MinimalDetails += $minimalObject
            $arrayToFile_AllDetails += $budget
        }    
    }
} #permission

$today = (Get-Date).ToString("MM-dd-yyyy")
$arrayToFile_AllDetails | export-csv -Path "C:tempAllBudget-AllDetails-$today.csv"
$arrayToFile_MinimalDetails | export-csv -Path "C:tempAllBudget-MinimalDetails-$today.csv"

Side note: make sure that you have c:temp or replace it with your path.

MinimalDetails csv file:

If this is what you were looking for, please accept the answer.
If you have any other questions, please feel free to ask.