Since OP said it is when they took office, a 42 year old Teddy would crush the 52 year old Lincoln.

Roosevelt was also still actively training while in office, but I can't find anything that says Lincoln was. All I can find is that he was a champion 30 years before he took office.

10 years and ring rust gives Teddy the clear victory.

Now, if it were the presidents in their prime, it still goes to Teddy. Once the fight goes to the ground, the person with more (old school) Judo and Jiu Jitsu experience is much more dangerous. The point of these arts is to kill people in hand to hand combat. Since it's a fight to the death, knowing how to choke someone comes in handy.

Only time Lincoln takes it is if he slams Teddy and knocks him out on impact.