Your situation may be somewhat related to this one, anonymous: http://bills-pokedex.tumblr.com/post/152882003486/i-adopted-my-chansey-after-she-trained-to-be-a
Or this one, even: http://bills-pokedex.tumblr.com/post/151956748386/i-tripped-down-the-stairs-yesterday-and-now-my
The difference here is that whereas one of the other pokémon was a former medical chansey and the other situation involved a minor injury, yours involves a newly adopted pokémon and a cold, which may be a bit trickier to handle and a bit more confusing to the pokémon, respectively. Still, a lot of the methods involved in both answers is the same.
The most important thing you can do for your blitzle right now is reassure him that what you’re feeling is completely normal (relatively speaking). Or, rather, that it’s easily curable. Tell him that you appreciate his help and concern but that all you need is rest and fluids and that he shouldn’t worry. Try to have your munna or any other pokémon (if you have any) comfort him and keep you occupied as you rest.
Whenever you can, though, try to maintain parts of your routine. If feeding your pokémon isn’t taxing on you, do so. If you can take them out for a walk or for training, do that as well. Getting back onto your normal routine will show your blitzle that all is essentially well and that he has nothing to worry about. Be sure to include as many of your other pokémon as possible in these routines so your blitzle can see how calm they are about the situation and gain a level of reassurance through his team.
But most importantly, take it easy and try to appear as relaxed as possible yourself. The more you worry—even about your blitzle, rather than yourself—the more he’ll pick up on that and interpret it as a reason to worry about you.
Best of luck, anonymous, and get well soon!