The minimum connection time for this connection to be valid is 31 minutes, so if you are lucky you will make the connection. If you’re delayed on the first train, since you didn’t allow enough time between the two trains, you might not be eligible for travel on a later train. However you seem to have an itinerary so if you show that you’re likely to be allowed to travel on a later train from King’s Cross without needing to buy a new ticket.
In the case of a valid interchange time, your rights are that you can use a later service with the same operator if your delay causes you to miss your train booked with an advance ticket, without paying more.
EDIT As stated in the following posts, if your interchange time is not “valid” but the tickets are bought in conjunction with an “invalid” itinerary, then you have the same right to use a later service in the case of delay to the first service, that’s to say without the need to buy a new ticket.