No, even if your company’s VAT-taxable turnover passes the threshold, you cannot charge VAT or include it on your invoices until you register for it and the HMRC assign you a VAT number.
However, you will have to pay VAT for the period from when the threshold is passed till you register for VAT. You can avoid losses by increasing selling prices after first informing your clients. As soon as your business gets a VAT number, you can include VAT on your invoices.