The link you provided is only in reference to VAT. The HMRC webpage does not discuss the presentation of these amount on either the Balance Sheet or Income Statement, only that for your reimbursable expenditures (payment that you make and then charge your customer for) you cannot claim any VAT paid as a tax credit, and you cannot charge VAT for those same amounts to your client.
For the set-up it would be best to use a Balance Sheet account, as that it is your intention to charge your customer for all of these amounts and not record and not have them factored into your net income calculation. Create an 'Other Current Asset' account named as 'Disbursements Receivable' for amounts you pay that you have yet to charge a customer. (Please double check with a UK CA as to whether or not this is the preferred treatment, they may still say to use either an income or expense account)
Most of the actual accounting work is best handled through the use of items and tax items/codes.
Here I am not familiar with UK VAT codes that may be available to you preset on UK QuickBooks. However, the idea is to create / use a sales tax item and sales tax code for your disbursement purchases so that you record the VAT you pay as an addition part of the cost of the disbursements you make. This could require creating a dummy vendor as a tax agency, in which you would leave the 'Tax tax on purchases separately to' blank, and creating duplicate of your sale tax item 'VAT on purchases' as VATD - VAT on Disbursements, and a new sale tax code "D" as Taxable linked to the sale tax item VATD for purchases (leave sales as blank)
For the disbursement purchases you make, you will need to create an Item in your Items List called 'Disbursements' as an other charge. You can use one or create as many needed with variations to the name (Materials, Fuel, & etc -Disbursements) to add detail to the 'Bills' you enter for supplier purchases or Invoices your create to customers. Check the item as being a reimbursable charge, you will reference the 'Disbursements Receivable' account for both Bills (purchase tax code D) and Invoices (tax code 'E' for exempt).
This way when you use the Enter Bill or Write cheques function to record disbursements you will use the 'Items' tab and select the 'Disbursements' items with the tax code D, so that the VAT you pay is calculated and entered as an additional part of the cost, and will not be claimed as a tax credit. (any other amounts not disbursements can be entered to the accounts on the expenses tab)
And when you Invoice your client, you will again use the same Disbursement items sales tax code E so that no VAT is charged on those items, VAT will only be calculated on the other sales/service you have provided items
