Taking on the existing tenants will not necessarily be problematic assuming that the tenants occupy the property on a standard Assured Shorthold Tenancy agreement.
You simply have to treat them as though they are potential, new tenants and ensure you carry out formal credit checks and obtain the necessary references. The vendor will have to inform the tenants of his intention to sell and serve them notice to quit. The tenants can then sign a new AST with you. Termination of the old AST and commencement of the new AST should happen simultaneously with the completion of the sale.
If the tenants are not on an AST, you may not be able to find a lender willing to give you a buy to let mortgage.