To go some way towards answering the question, firstly, no, the movements must not all be the same tempo - far from it.
Aside from the Preludes, which despite some dance rhythms are not dances,
i. Allemande - fundamentally stately. Usually played too fast IMO - don't be fooled by the fast note values. Think of a floating 4 in the bar, decorous, a little pompous.
ii. Courante - means Running in French. Think bustling rather than rushing. Texture is often broken chords. Btw a Corrente is the Italian equivalent, is usually a little faster and lighter of texture.
iii. Saraband - the turncoat of the suite, when first danced in Europe was fast and lascivious. In Baroque is very slow (though still a dance!) can be very cerebral, e.g. Suite 5, or sensual. Extra stresses often on beat 3 or 2.
iv. the "galanteries" e.g. Menuets, Bourrées, Gavottes, can follow the Saraband. Minuet/Menuet is a little rustic, the Menuet 2 often more notably so, with implied drones like bagpipes. Tempo is delicate, the dance has an 'up-step' on beat 1. Bourrées often more rough, can be a little quicker, Gavottes a bit the same.
v. Gigue - means Jig, the fastest and most energetic movement.
The point overall is to vary the tempi across the suite so that both the national-like implied character is present, and so that the whole makes an agreeable contrast and arc from beginning to end. In ascending order of tempo for the main four dances its Saraband, Allemande, Courante, Gigue, so one can see how the suite manages intensity, variety, and a satisfyingly exciting finale. Preludes and Galanteries complicate everything. A prelude almost by definition needs to be considered in isolation as a piece but also in context of the suite that follows. In probably all cases it will be faster than the Allemande. Menuets etc basically fit in with what's on either side, at the very least, even a fairly excitable Bourrée should never over-shadow the Gigue that will follow it.
So if one is playing only some of the movements, what to do? The issue of balancing tempo and intensity across the whole suite is not present, but still the fundamental character of the dance should ideally be present.
Simon Ambridge Series 40 (2005)
Trevor Semple Series 88 (1992)
Louis Panormo (1838)
Alexander Batov Baroque Guitar (2013)
Simon Ambridge 'Hauser' (2018)