Almost every carriage on Avanti trains has luggage storage at the ends of the carriages as well as above the seats. I think the only exception might be the carriage with the shop, on some trains. Depending on the type of train, there might also be luggage storage in the middle of the carriage. The luggage storage areas in Standard class can get quite full, though, due to the seating density, and you'd have a better chance of finding space in the Standard Premium carriage, which, in my experience, is usually fairly quiet. Standard Premium (which is a First Class carriage without the "free" food and drink) has no airline-style seating, and you can also store luggage between the seatbacks.

(It's not necessary to pay in advance for Standard Premium seating: you can just take a seat on the train and pay the train manager when he comes to check tickets.)