I don't think that there is a specified total annual budget for the Royal Navy per se, in fact, I don't think that there is a specified total annual budget for the Armed Forces as a whole - expenditure on the Armed Forces is expected to come from the funds allocated to the MoD via the Defence Budget.
It would be reasonable to equate the Defence Budget with the Departmental Expenditure Limit (DEL) allocated to the Ministry of Defence by Parliament in any given financial year.
The MoD's DEL for the financial year 2011/2012 was £33,750 Million, IIRC.
Have a browse through the most recent Annual Report for the MoD; it includes the Department's accounts:
Having said what I have said, there seems to be a figure on page 120 of last year's MoD Annual Report which might fit the bill from your point of view ...
Look at the line for Navy Command in the table on that page and see what you come up with.
The MoD's Annual Report for the financial year 2011/2012 should become available in the next few weeks, by the way. If you look at that, you should be able to see an even more current figure for Navy Command in the equivalent table to the one I have referred to in the 2010/2011 Report.