This is a serious concern and worth pursuing directly with the facility, but how you approach that conversation matters as much as what you ask.
Facilities are legally required to provide inmates with necessary medical care, and failing to treat a documented chronic condition like hypertension or high cholesterol can rise to the level of a constitutional violation under the Eighth Amendment. That is the legal backdrop. In practice, medication gaps happen for several reasons. A prescription may not have transferred properly from the county jail or previous facility. The facility's medical department may not have received his complete medical history. There could be a formulary issue where the specific medication he was on is not stocked and a substitute needs to be approved. Or it could simply be an administrative gap that nobody has flagged yet.
The most effective way to get this resolved is to call the facility and ask to speak with a lieutenant, counselor, or the medical department directly. When you get someone on the line, be calm, polite, and specific. Explain that your husband has documented diagnoses of high blood pressure and high cholesterol, that he was on specific medications before his transfer, and that he appears not to be receiving them. Ask whether they are aware of this and what the process is for ensuring his prescriptions are continued.
The tone of that conversation genuinely matters. Staff who feel respected and approached as professionals will go out of their way to help. Staff who feel they are being accused or demanded of will do the minimum and move on. Please and thank you are not just pleasantries here. They are strategy.
On his end, your husband should submit a written medical request through the facility's sick call process specifically asking to see medical staff about his ongoing prescriptions. A written request creates a paper trail that an oral conversation does not, and that documentation becomes important if the situation is not resolved promptly.