I want to encode two large integers of possibly different maximum bit lengths into a single integer. The first integer is signed (can be negative) whereas the second is unsigned (always non-negative). If the bit lengths are m and n respectively, the bit length of the returned integer should be less than or equal to m + n.

Just n (but not m) is known in advance and is fixed. The solution will as an example be used to combine a signed nanosecond timestamp of 61+ bits along with 256 bits of unsigned randomness to form a signed 317+ bit unique identifier.

I'm using the latest Python. There is a related preexisting question which addresses this in the special case when m == n.

Since n is fixed, the problem is trivial: Encode (a, b) as a•2ⁿ+b.

If m and n were not fixed, the problem is impossible because it asks us to encode both (two-bit a, one-bit b) and (one-bit a, two-bit b) in three bits, which means we must encode the twelve possibilities (0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 1), (3, 0), and (3, 1) in the eight combinations of three bits, which is impossible.