Partial answer while I search for a copy of De Vaan…
These two words were distinct in Old Latin. The word for "wide" was originally stlātus, while "borne" was tlātus.
Stlātus, according to Lewis and Short, is connected to Sanskrit strnāmi, Greek stórnymi and stratós, Latin sternō and stratus, and English strew. The st- is still visible in the archaic word stlāt(t)a, "barge". All of these come from PIE *sterh₃- "extend, stretch". Unfortunately, L&S don't specify how the *r turned into l, but d turning into l is well-attested (see lacrima, lingua), and this might be a similar process?
Tlātus, on the other hand, was originally the fourth principal part of tollō, tollere, tulī, tlātus, cognate with Greek tlētos "steadfast". (In Classical times the third and fourth parts were stolen by the unrelated verb ferō, so tollō needed to add prefixes to keep them distinct.) This verb came from the PIE root *telh₂ "bear". The varying vowels go back to the patterns of PIE ablaut: tlātus in particular is from the zero-grade *tlh₂, with the laryngeal turning into the ā, while the other parts come from a full grade. The present stem gets its extra l from a nasal infix that assimilated.
Both of these words then got simplified when Latin stopped allowing the combination tl, and ended up looking the same by pure coincidence. As Alex B points out in his own answer, this is a known sound change: locus comes from stlocus.