If your subcontractor is VAT registered, then they need to charge VAT to you (NB there exceptions for a few very specific types of services, but I am assuming these don't apply).
As you are not registered, you cannot reclaim that VAT, so it increases your normal business costs. These costs will reduce your taxable profits for corporation tax.
If the expense is very specific to the work you are performing and can be passed on to your client as a "disbursement" based on your contractual engagement with the client (and your client is VAT registered) then the client may be able to reclaim the VAT. Disbursement rules are complex, but ultimately your client would be responsible for ascertaining whether they apply. You would still charge the VAT inclusive amount when recharging a "disbursement".
In reality the disbursement option is likely to tick the "not relevant" or "too difficult" box.
If won't effect your profitability, but may be worth looking into if the subcontract fees are very high on specific client jobs, if it helps attract work.
Finally, the inability to reclaim your subcontractor's VAT doesn't necessarily mean that you should rush out to become VAT registered yourself if you are under the VAT income threshold, as there are a number of factors you should consider before reaching that decision.
However, the level of VAT embedded in your business costs is one of those factors, so it may be time to perform another review of whether VAT registration is right for you.