As Piggling indicates, your charity is bound by its constitution.
If, for example, such a constitution was to state that the aim of the charity was solely to fund medical care for animals, you could lawfully give money to another charity with similar aims (e.g. the PDSA). You could also give money to a relevant specific project (e.g. the creation of an animal hospital) set up by the RSPCA but you couldn't contribute to the RSPCA's general funds (as they do much more than just fund the medical care of animals). You certainly couldn't give money to Oxfam or to the RNLI.
The essential test is embodied in this question:
"Is the money received from donors being put to the use which they thought that it would be when they agreed to donate it (either directly or, for example, by attending fund-raising events)?"
If the answer is "Yes", you're in the clear.
Chris