BBC listed 'Can I get a...' as the most irksome Americanism. Then what is the most British way of making an order?
Firstly, let me point out that is should be written as
Can I getta...?
which is asking about the possibility of having something, possibly in a particular way.
The BrE equivalents might be
Please could I have...
Could I please have...
I would like to have...
Would it be possible to have... if it's not too much trouble.
AmE Can I getta pint of beer, (please)?
BrE (A) lager, please (pint is understood as the default in any pub)
AmE Can I get that on a roll?
BrE Please could I have that on a bap.
AmE Can I getta hamburger without pickles and lettuce?
BrE Would it be possible to have the hamburger without pickles and salad?
if the hamburger has already been order then Sorry is prefixed to the request
BrE Sorry, would it be possible to have the hamburger...
AmE Can I getta train there?
BrE Would you know if the overland or tube goes there?