Is it standard British usage to add forms of the verb "Do" after a conditional like "Would"? e.g., If I could, I would do"?
What is going on here is not the addition of 'do', and has little to do with conditionals in particular. It presents a different prioritization of reduction rules.
There is no added 'do'. The construction in your example is that 'would', for instance, takes another verb, and that in backward references the phrase would therefore properly end with 'would do so'. Nor does the auxiliary verb have to be 'do'.
AmE more aggressively suppresses auxiliary verbs for which the objects have already been suppressed.
He has done that, and I would do so, too.
He pays his rent because he has to do so.
He will be dead before you are so.
You fill out tax forms faster than I do so.
becomes these in AmE.
He has done that and I would, too.
He does that because he has to.
He will be dead before you.
You fill out tax forms faster than I.
In BrE, you are more likely to remove the object of the verb, without removing the verb itself.
The 'exception that proves the rule' is the optional hanging verb in the last two examples. When the omitted part is less surely guaranteed to be a simple back-reference like 'so' or 'that', it is sometimes slightly clearer to hint at the omission. So sometimes these examples have their terminal auxiliary verbs. But it sounds more well-considered when they are omitted as well.