The toad's grapple on you ends, yours on it does not by RAW, but the DM should probably adjudicate that
Being swallowed is not a condition, but it is described with its own effects, here from the Giant Toad (but the language is essentially the same for the Behir, the Remorhaz, the Tarrasque, and the Giant Frog, except that some of them do not need to grapple first, and except that the damage differs in amount):
If the attack hits, the target is swallowed, and the grapple ends. The swallowed target is blinded and restrained, it has total cover against attacks and other effects outside the toad, and it takes 10 (3d6) acid damage at the start of each of the toad's turns.
This tells you that when you are swallowed, the grapple of the toad grappling you from its bite attack ends. Instead, you are blinded and restrained, and have total cover. Restrained among other effects has the same functional effect as the first bullet of grappeled: A restrained creature’s speed becomes 0, and it can’t benefit from any bonus to its speed., just as if it was grappled, so for practial purposes of movement, there is no change for the swallowed creature.
Does your grapple on the toad end?
The Grappled condition (p. 290, PHB) says:
- A grappled creature’s speed becomes 0, and it can't
benefit from any bonus to its speed.
- The condition ends if the grappler is incapacitated
- The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.
You are not incapacitated by being swallowed. Is the toad still within your reach? Reach is not formally defined, but the combat rules on p. 195, PHB tell us:
Most creatures have a 5-foot reach and can thus attack targets within 5 feet of them when making a melee attack.
Reach is described as the distance within which a creature can attack
a target with a melee attack. While the toad cannot attack you any more, you can attack the toad, from the inside, so it is still within your reach, and none of the conditions ending the grapple is fullfilled. So, strictly as written, when you are inside the toad, the toad is still grappled.
Adjudicating the rules
If the toad is grappled, it's speed is reduced to 0 by the condition. However, you have no leverage point against which to stop the toad from moving, so based on common sense, this outcome is nonsensical. (For example, in the "Never give up" image in Dale's answer, the stork could still move.) This would be an area where it would not be surprising for the DM to overrule the mechancial effect, and allow the toad to move with you inside.