Do HEMA or rapier skills carry over to Olympic fencing, and does Olympic fencing help in HEMA?
Some of it carries over, but not in the way most people hope: specific HEMA techniques rarely transfer to Olympic fencing, while the fundamentals of Olympic fencing, footwork, distance and timing, transfer very well to HEMA. Both come from the same family of thrusting-sword fencing, and a lot of the thinking is shared, such as hitting with opposition so you are covered while you score. The difference is that each is now played under its own rules with different weapons, and techniques survive where the rules reward them.
From rapier into épée, the closest Olympic cousin, the mindset fits well. Épée has no right of way, the whole body is target, and a double touch scores for both fencers, so a rapierist's habit of not getting hit while hitting is valuable. Much of the rest does not fit. Épée scores with the point only, so cuts count for nothing. Grabbing, using the free hand to block, jostling and turning your back are all penalized. Circling footwork has no room on a strip 1.5 to 2 meters wide, and stepping off the side stops the bout and costs you a meter of ground. An unusual move may surprise someone once or twice before they adjust.
Going the other way, Olympic fencers usually arrive in HEMA with sharp footwork, distance control and fast, precise point work, which are real advantages. The adjustments are the weapon and the rules. A practice rapier commonly weighs over a kilogram and has a longer blade, while a competition épée must weigh under 770 grams, so the tempo is slower and committing to an action takes more. HEMA rulesets vary by event, and many judge hits by eye and punish double hits much harder than épée does, so a touch that wins in épée may score poorly.
Plenty of people enjoy both, and neither is the "real" one: Olympic fencing is a sport refined for competition, and HEMA reconstructs how historical masters taught people to fight. If you do both, tell each coach, keep the two styles separate in your head for the first months, and expect each to make you a little worse at the other before it makes you better. Fencing someone from the other discipline is great fun if you leave your ego at the door. Our swordplay page lists Texas HEMA and historical rapier groups.