The next brew on tap, an oatmeal stout, calls for British ale yeast.

My house strain (thanks to washing) is Wyeast Pacman.

Will there be a significant difference if I ferment with Pacman vs. a British ale yeast?

Which British yeast? If nottingam then probably not much difference since it's pretty clean. If you're wanting a British ester profile or lower attenuation then that'll show up as lacking with pacman. Depends on what kind of stout you're after.

I am just about to bottle an oatmeal stout I did with Pacman and I have to say it's much cleaner than any beer I've done with an english strain. I've really only used 002 so I don't know what the cleaner strain (007) would do.
I think the choice depends on which way the recipe leans. Mine is all roast and late hops so I like the cleaner fermentation profile. If it's a more balanced recipe, then the english yeast character will be really nice.

Good luck and I don't think you can go wrong.

I know it's sort of a necrobump, but I thought I'd chime in: I've brewed an oatmeal stout (mcquakers' from BCS) using pacman, because that's what I had. It was fabulous- a bit dryer than when I brew the same recipe with 002, but the malt flavors came through nicely.

I may do the Anderson Valley Barney Flats Oatmeal Stout recipe in this months BYO. It calls for a British Ale yeast and again, all I have is Pacman.

In an oatmeal stout, I think you can use a clean ale strain like pacman and be totally fine. You won't get the subtle fruity esters that a British strain at moderate temps can provide, but it will still make a fine beer. I'd pitch the pacman without any hesitation. Rogue Shakespeare is an oatmeal stout fermented with pacman of course, and everyone knows that beer is fantastic.