You may planeswalk at the time TARDIS's triggered ability resolves, not at the time you cast the next spell.

You may planeswalk only in games using the Planechase variant. If you are instructed to planeswalk in any other game, nothing happens.

A spell's mana value is determined only by its mana cost. Ignore any alternative costs, additional costs, cost increases, or cost reductions.

Cascade triggers when you cast the spell, meaning that it resolves before that spell. If you end up casting the exiled card, it will go on the stack above the spell with cascade.

When the cascade ability resolves, you must exile cards. The only optional part of the ability is whether or not you cast the last card exiled.

If a spell with cascade is countered, the cascade ability will still resolve normally.

You exile the cards face up. All players will be able to see them.

If you cast a card "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those to cast the card.

TARDIS : règles, fonctionnement et interactions (MTG)

TARDIS is one of those cards players keep asking about. Here is, in plain words, what its official text says, the rulings published by Wizards of the Coast, and the combos players actually run.

Official card text

Flying Whenever this Vehicle attacks, if you control a Time Lord, the next spell you cast this turn has cascade and you may planeswalk. Crew 2 (Tap any number of creatures you control with total power 2 or more: This Vehicle becomes an artifact creature until end of turn.)

How it works : TARDIS

Original English text — Wizards of the Coast / Scryfall

#1October 13, 2023· Wizards of the Coast
You may planeswalk at the time TARDIS's triggered ability resolves, not at the time you cast the next spell.
No other card is named in this ruling.
#2October 13, 2023· Wizards of the Coast
You may planeswalk only in games using the Planechase variant. If you are instructed to planeswalk in any other game, nothing happens.
No other card is named in this ruling.
#3October 13, 2023· Wizards of the Coast
A spell's mana value is determined only by its mana cost. Ignore any alternative costs, additional costs, cost increases, or cost reductions.
No other card is named in this ruling.
#4October 13, 2023· Wizards of the Coast
Cascade triggers when you cast the spell, meaning that it resolves before that spell. If you end up casting the exiled card, it will go on the stack above the spell with cascade.
No other card is named in this ruling.
#5October 13, 2023· Wizards of the Coast
When the cascade ability resolves, you must exile cards. The only optional part of the ability is whether or not you cast the last card exiled.
No other card is named in this ruling.
#6October 13, 2023· Wizards of the Coast
If a spell with cascade is countered, the cascade ability will still resolve normally.
No other card is named in this ruling.
#7October 13, 2023· Wizards of the Coast
You exile the cards face up. All players will be able to see them.
No other card is named in this ruling.
#8October 13, 2023· Wizards of the Coast
If you cast a card "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, you must pay those to cast the card.
No other card is named in this ruling.
#9October 13, 2023· Wizards of the Coast
If the card has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.
No other card is named in this ruling.
#10October 13, 2023· Wizards of the Coast
Due to a 2021 rules change to cascade, not only do you stop exiling cards if you exile a nonland card with lesser mana value than the spell with cascade, but the resulting spell you cast must also have lesser mana value. Previously, in cases where a card's mana value differed from the resulting spell, such as with some modal double-faced cards or cards with an Adventure, you could cast a spell with a higher mana value than the exiled card.
No other card is named in this ruling.
#11October 13, 2023· Wizards of the Coast
The mana value of a split card is determined by the combined mana cost of its two halves. If cascade allows you to cast a split card, you may cast either half but not both halves.
No other card is named in this ruling.

Interactions with other cards

These cards' text mentions TARDIS.

The Thirteenth Doctor

Paradox — Whenever you cast a spell from anywhere other than your hand, put a +1/+1 counter on target creature. Team TARDIS — At the beginning of your end step, untap each creature you control with a counter on it.

The Ninth Doctor

Haste Into the TARDIS — Whenever The Ninth Doctor becomes untapped during your untap step, you get an additional upkeep step after this step.

The First Doctor

When The First Doctor enters, search your library and/or graveyard for a card named TARDIS, reveal it, and put it into your hand. If you search your library this way, shuffle. Whenever you cast a spell with cascade, put a +1/+1 counter on target artifact or creature.