Try using a derived table instead.

SELECT p.PatientID, p.PatientType, p.AccountNumber,
    p.FirstName + ' ' + p.LastName PatientFullName,
    p.CreatedDate, DATEDIFF(hour, p.CreatedDate, GETDATE()) TotalTime,
    bg.AvgBG 
FROM Patients p 
JOIN (SELECT PatientID, AVG(BGValue) AvgBG
    FROM BloodGlucose group by PatientID) BG
ON BG.PatientID = p.PatientID 
WHERE AvgBG > 60;

Derived tables work in sets and correlated subqueries work row-by-agonizing-row, that is why mine is faster.

The aliases in the WHERE clause can only come from the FROM clause. Here is a way to rewrite your query:

SELECT p.PatientID, p.PatientType, p.AccountNumber,
       p.FirstName + ' ' + p.LastName as PatientFullName, p.CreatedDate,
       DATEDIFF(hour, p.CreatedDate, GETDATE()) TotalTime,
       av.AvgBG
FROM Patients p join
     (SELECT PatientId, AVG(BGValue) as AvgBG
      FROM BloodGlucose
      group by PatientID
     ) av
     on p.PatientId = av.PatientId
WHERE av.AvgBG > 60;

SELECT p.PatientID, p.PatientType, p.AccountNumber, p.FirstName + ' ' + p.LastName PatientFullName, p.CreatedDate, DATEDIFF(hour, p.CreatedDate, GETDATE()) TotalTime, bg.AvgBG  
FROM Patients p  
outer apply (SELECT PatientID, AVG(BGValue) AvgBG   
FROM BloodGlucose where PatientID = p.PatientID  
 group by PatientID ) BG 
 WHERE AvgBG > 60;

this should also work pretty quickly

SELECT p.PatientID, p.PatientType, p.AccountNumber, p.FirstName p.LastName PatientFullName, p.CreatedDate,
DATEDIFF(hour, p.CreatedDate, GETDATE()) TotalTime, a.AvgBG
FROM Patients p,
(SELECT PatientID, AVG(BGValue) AvgBG
  FROM BloodGlucose
group by PatientID
having AVG(BGValue) > 60) a
where p.PatientID = a.PatientID

You can try to use WITH clause in standard SQL.

WITH Subquery AS
(
    Your Code
)

And then your code

SELECT ....
Subquery As sq
From Patients p
WHERE(subquery)>60

Use a common table expression:

;WITH CTE (PatientID, PatientType, AccountNumber, PatientFullName, CreatedDate, TotalTime, AvgBG) AS (
    SELECT p.PatientID, p.PatientType, p.AccountNumber, p.FirstName + ' ' + p.LastName PatientFullName, p.CreatedDate,
    DATEDIFF(hour, p.CreatedDate, GETDATE()) TotalTime,
    (SELECT AVG(BGValue) FROM BloodGlucose WHERE PatientID = p.PatientID) AvgBG
    FROM Patients p
)

SELECT PatientID, PatientType, AccountNumber, PatientFullName, CreatedDate, TotalTime, AvgBG
FROM CTE
WHERE AvgBG > 60