The regiment later merged with two others from Yorkshire, becoming the Yorkshire Regiment, and Sir Tom was made an honorary colonel last August.
A bugler sounded The Last Post at the end of the private service.
Six representatives from the Army Foundation College in Harrogate, where Sir Tom was made an honorary colonel, also formed a ceremonial guard.
There are plans to plant trees around the world in his honour, with Ms Teixeira hoping the Trees for Tom initiative will result in a wood in his home county of Yorkshire and the reforestation of part of India, where he served during the Second World War.
Sir Tom asked that his epitaph reads: “I told you I was old”, in reference to comedian Spike Milligan’s famous epitaph: “I told you I was ill.”
Once Covid-19 restrictions permit, the family will inter Sir Tom’s ashes in Yorkshire, with his parents and grandparents in the Moore family plot.