A lender can adjust their SVR any time they want to.
However, if your contract states (as you have written) that the rate you will pass onto will be at 2.00% above bank of england base rate then this is a written statement which specifies the rate you will transfer onto (2% above BoE BR).
If the SVR is written as a "A variable rate which is currenly xyz%" then it allows the lender room to adjust this rate to what ever they want, when ever they want. They have not tied themselves to a specific rate, only that the rate is variable.
read through your formal mortgage offer and see how it is termed as this could give you insight as tyo what may happen. Just because a company becomes nationalised does not mean the costs will automatically go up.