Some whistles are made to be tunable. Typically, this means that the mouthpiece can slide up and down the shaft (Is it getting a little warm in here?) Others aren't made to be tunable, but can be modified to be made tunable. I fix almost all of mine to be what I call sorta-tunable. You might want to try this. DO UNDERSTAND THAT IF YOU ATTEMPT TO MAKE A TINWHISTLE TUNABLE, SOME RISK OF SCREWING UP THE WHISTLE EXISTS. DON'T TRY ANY OF THESE METHODS UNLESS YOU CAN TOLERATE RUINING THE WHISTLE IF YOU MAKE A MISTAKE.
Sorta-tunable means that you can make some adjustment in the overall pitch of the instrument by sliding the mouthpiece up and down. It's not as effective as a good, professionally made tuning slide.
This method involves applying heat to the mouthpiece where it is glued to the shaft. Most people do this with hot water (not boiling). It requires some patience. You run the hot water over the place where the mouthpiece and shaft overlap. Then you pull them apart. It often takes several tries. I hold the mouthpiece and the shaft with dishtowels to keep from burning myself. If I have trouble keeping a grip on the shaft, I wrap it with plastic wrap which improves the friction situation. (HEY! Stop giggling out there!) In many cases, the heat will melt the glue and allow the mouthpiece to slide off. You can then lubricate the shaft (It really IS getting warm in here) and replace the mouthpiece.
I think for most cheap whistles, however, this allows only for making the pitch flatter. This is because most good manufacturers have tuned the whistle so that it is in tune when the mouthpiece is all the way down. In theory, then, if you wanted to make it possible to sharpen the pitch relative to the perfect bell tone, you would have to shorten the shaft slightly. I could imagine taking off a little less than a centimeter, on the mouthpiece end, either by cutting with a hacksaw or by grinding down the shaft. I have never done this and so therefore don't recommend it. However, I've heard from people who do this.
One tuning anecdote. Have you ever seen the televised concert of Roger Daltry doing Who songs? The Chieftains play as a guest and the tinwhistle player, Paddy Moloney, has a huge piece of duct tape kind of crudely wrapped around the area where the mouthpiece joins the shaft. It occurred to me later that this is to keep the tuning slide from slipping. It's pretty comical looking. Someone emailed me and set me straight and said that Paddy was making fun of Roger Daltry, who often has to add duct tape to his mic cords because of the way he slings them around while performing.