Up front, I am a relative newbie to the whistle. I have a tunable Sweet whistle that has a slightly loose head; the head pulls up just a bit too easily if I am not careful There is some sort of sealing material (cork?) on the male part of the whistle that, if just a slight tad thicker, would keep this from happening. I read about conditioning solutions/emulsions for whistles, but I don’t want to mess something up beyond repair. What suggestions do you have to help with this loose head piece? 
If the cork is worn, maybe a bit of PTFE plumbers tape, just a sliver, else get it re corked.
Check out what kind of material is used for the joint. R. Sweet often used felt. Especially in the older instruments.
Cork & thread are also common tenon materials as are rubber O rings.
As for what to do, you can have a go at fixing it yourself. There was a recent thread about a DIY recorking. Thread is the easiest (and I think best) option because you can easily determine how much thread to use and thus adjust the joint tension to your preference. Lasts a long time too. If you use cork or felt you have to be careful of what glue you use. (You may want to recork the instrument ten years from now and scraping super glue off the wood isn’t going to be fun!
Taking it to a woodwind repair shop is a cheap and easy alternative and it’ll be done right.
If the existing cork has simply compressed, it can be revived by heat. Here’s a thread from the Flute forum about revitalizing a head cork. Same principal works with tenon cork. Heat (carefully) with steam from a kettle—or you could use an open flame (making very sure that you are only heating the cork, not incinerating it). Perhaps try a bit of heat and see if it has any effect. Then try again until the cork is again snug. If that doesn’t work, you can try the plumbers’ tape as a short term fix or getting it re-corked by an instrument repair person/technician.
Best wishes.
Steve