The answer to your question depends on if you treat the dwarf as a mount. If you do, the mounted combat rules cover you.
P. 198 PHB:
While you're mounted, you have two options.You can either control the mount or allow it to act independently. Intelligent creatures, such as dragons, act independently. […]
An independent mount retains its place in the initiative order. Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes.
Since the dwarf is intelligent, he gets his own initiative order and acts independently, as he wishes on his own initiatve. The halfling will move around with him, instead of using his own movement, until he dismounts. He will get to otherwise act or attack on his own initiative. Both can attack independently on their respective turns, and suffer no disadvantage.
Normally, a dwarf would not be able to act as a mount, because:
A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount, using the following rules.
While its up to the DM what "an appropriate anatomy" is, from the example mounts in the PHB it seems a bit of a stretch to allow a humanoid body plan for a mount that is just one category larger.
If you do not want to treat the dwarf as normal mount and want to impose additional restrictions such as disadvantage to the attacks, because they must hold up and cling on to each other, that is entirely up to you: homebrew rules, like allowing riding a creature that is not a mount, require hombrew solutions. For example you could declare this works like a normal intelligent mount, but with disadvantage to attacks — it's up to you.
PS. Note that even if the players are acting at different initiative, the are still all acting "at the same time", because initiative is just an artifact for the players. In game, everyone is acting during the same six-second round simultaneously. (Thanks, @Jack)