The short answer is: technically, no. The weaknesses of MD5 are not an issue here. However MD5 is seriously inappropriate, for it is the wrong king of security primitive; also its reputation is tarnished.
If a collision attack was to be feared, then using MD5 would be a disaster, for it is now hopelessly broken w.r.t. to collision resistance; but that does not seem to be a serious threat in the stated usage. At worse, it seem possible to exhibit different (master password, nickname) pairs resulting in the same account password, but I fail to see that it leads to a practical attack.
The main technical drawback of using MD5 in the stated usage is, paradoxically, MD5's main quality: it is fast. That makes password cracking much easier than for a purposely slow Password-Based Key Derivation Function like PBKDF2 or Scrypt, and slightly easier than if using a hash stronger than MD5, like one of the SHA-2. However being fast is not considered an MD5 weaknesses; using a fast hash is a weakness of the proposed usage.
Also, from a Public Relation standpoint, using MD5 is a bad idea, much like using a tire known to explode at high speed is a bad idea for a manufacturer of low speed cars.
Update: Unless the master password is imposed on the user (which makes the scheme much less useful), a controllably slow Key Derivation Function is required, and as an aside should be followed by a better method to generate an acceptable password (as pointed in that other answer). I recommend Scrypt, because it uses controllably much RAM and (possibly) CPU cores, hence raising the cost of a determined attack for a given penalty to the user (a small parameterizable delay, and using the computer's resources during that time). See this table from the paper defining Scrypt:
Note: A letter is random among 26 lowercases; a character is random among 95 ASCII symbols; text entropy per NIST SP800-63-1.