Convert an ASCII art table to a UTF-8 table

Python 3, 914 898 827 823 594 587 569 540 469 bytes

Edit: significantly changed strategy, now making a bitfield of neighbors (similar to dead-possum's answer). I've left the earlier version below.

H='─│═-|=└╘++++┌╒├╞++┘╛++┴╧┐╕┤╡┬╤┼╪'
def n(l):
 def j(r,c,t=0):O=(0,r-1,c),(1,r,c+1),(2,r+1,c),(3,r,c-1);v=f(r,c);b=t|any(f(Y,X)=='='for i,Y,X in O);l[r][c]={'+':H[b+2*sum((f(Y,X)in H)<<x for x,Y,X in O)],**dict(zip(H[3:6],H))}.get(v,v);[f(Y,X)!=';'and v in'+++-|='[i%2::2]and j(Y,X,v=='=')for i,Y,X in O]
 for i,I in enumerate(l):
  if'+'in I:f=lambda r,c:l[r][c]if len(l)>r>=0and 0<=c<len(l[r])else';';j(i,I.index('+'));break

Try it online!

Input is in the form of a list of lists of characters, which is modified in place. Recurses from the first + that it finds.

x=range
C='┌┐└┘','╒╕╘╛'
D='┬┤┴├','╤╡╧╞'
A='┼╪'
H,V,T='─│═'
J={'-':H,'|':V,'=':T}
K=C[1]+D[1]+A[1]+'='+T
E=('+|','+-=')*2
F=['+|'+V,'+-='+H+T]*2
O=(0,-1,0),(1,0,1),(2,1,0),(3,0,-1)
for i in x(4):
 for j in{0,1,2,3}-{i}:F[i+2&3]+=D[0][j]+D[1][j]
 h=C[0][i]+C[1][i];F[i&2]+=h;F[3-2*(i&1)]+=h
def n(l):
 for i,I in enumerate(l):
  if'+'in I:r=i;c=I.index('+');break
 else:return l
 def f(r,c):
  try:assert c>=0 and r>=0;return l[r][c]
  except:return'\0'
 def j(r,c):
  v=f(r,c)
  l[r][c]=J.get(v,v)
  if v=='+':
   X=[f(r+Y,c+X)for i,Y,X in O];B=any(x in K for x in X);X=[X[x]in F[x]for x in x(4)];L=sum(X)
   if L in(2,3,4):l[r][c]=D[B][X.index(False)]if L==3 else C[B][X[0]*2+X[3]]if L==2 else A[B]
  for i,Y,X in O:
   if v in E[i]and f(r+Y,c+X)in E[i]:j(r+Y,c+X)
 j(r,c);return l

Try it online!

Here's the closest thing I have to an ungolfed version:

def tr(s):
    t='┌┐└┘','╒╕╘╛'
    t2='┬┤┴├','╤╡╧╞'
    A = '┼','╪'
    H,V,T = '─│═'
    Th = ''.join(x[1]for x in (t,t2,A))+'='+T
    ps = ['+|'+V, '+-='+H+T, '+|'+V, '+-='+H+T]
    ps2 = ('+|', '+-=')*2
    for i in range(4):
        for j in {0,1,2,3}-{i}:
            ps[(i+2)%4] += t2[0][j]+t2[1][j]
        h=t[0][i] + t[1][i]
        ps[i & 2] += h
        ps[3 - 2 * (i & 1)] += h

    l = [list(x) for x in s.split('\n')]
    r = 0
    for i,I in enumerate(l):
        if'+'in I:
            r=i;c=I.index('+')
            break
    def g(r,c): return l[r][c]
    def G(r,c):
        if r >= 0 and r < len(l) and c >= 0 and c < len(l[r]):
            return g(r,c)
        return '\0'
    def process(r,c):
        v = g(r,c)
        if v == '-': l[r][c] = H
        elif v == '|': l[r][c] = V
        elif v == '=': l[r][c] = T
        elif v == '+':
            all=[G(r-1,c),G(r,c+1),G(r+1,c),G(r,c-1)]
            bold=any(x in Th for x in all)
            for i in range(4):all[i] = all[i] in ps[i]
            N,E,S,W=all
            tt=sum(all)
            if tt == 3:
                l[r][c]=t2[bold][all.index(False)]
            elif tt == 2:
                l[r][c]=t[bold][N*2+W]
            elif tt == 4:
                l[r][c]=A[bold]
            else: return
        for i,(dy,dx) in enumerate(((-1,0),(0,1),(1,0),(0,-1))):
            if v in ps2[i] and G(r+dy,c+dx) in ps2[i]:
                process(r+dy,c+dx)
    process(r,c)
    return l

Python 3, 392 281 bytes

Golfed it quite a bit more and converted to a recursive solution instead of an iterative one:

def h(a):
 def g(i):
  k=-3;d=a[i]=='=';z[i]=''
  for p,j,r in zip((1,2,4,8),(i+1,i+w,i-1,i-w),('+-=','+|')*2):
   if 0<=j<len(a)and{a[i],a[j]}<={*r}:k+=p;d|=a[j]=='=';z[j]and g(j)
  z[i]="┌╒!!─═┐╕┬╤@@└╘││├╞┘╛┴╧┤╡┼╪"[2*k+d]
 w=a.find('\n')+1;z=[*a];g(a.find('+'))
 return''.join(z)

Takes a string of equal length rows separated by newlines, and returns a string in the same format. May throw an exception on invalid input.

Previous solution:

def h(a):
 i=a.find('+');q=[i];w=a.find('\n')+1;z=[*a]
 while q:
  i=q.pop();c=a[i];k=-5
  for p,j in enumerate((i+1,i-1,i+w,i-w)):
   r='++-|='[p>1::2]
   if 0<=j<len(a)and a[i]in r and a[j]in r:
    k+=1<<p;q+=[j][:z[j]<'─']
  z[i]='│'if c>'='else'─═'[a[i]>'-']if c>'+'else"┌╒┐╕┬╤@@└╘┘╛┴╧##├╞┤╡┼╪$$"['='in a[abs(i-1):i+2]::2][k]
 return''.join(z)

Ungolfed version:

def h(a):
    i = a.find('+')         # find index of first '+'. It is first node
    q = [i]                 # in the queue of indexes to convert to unicode
    w = a.find('\n')+1      # width of the table
    z = [*a]                # strings are immutable, so copy it to a list

    while q:                # while the queue isn't empty
        i=q.pop()           # get the next index to process
        c=a[i]              # and the associated character

        k=-5                # 'k' is the index into the unicode string, U.  The way they
                            # are encoded, the first unicode value is at index 5. 

                 # directions  E   W   S   N
        for p,j in enumerate((i+1,i-1,i+w,i-w)):  # j is the index of an adjacent cell

            # r='++-|='[p>1::2] is equivalent to:
            if p > 1:
                r = '+|'    # compatible symbols for vertical connections
            else:
                r = '+-='   # compatible symbols for horizontal connections

            # if adjacent cell index is valid and the characters are compatible
            if 0 <= j < len(a) and a[i] in r and a[j] in r:
                k += 1<<p                 # update the unicode symbol index

                # q += [j][:z[j]<'─'] is equivalent to:
                if z[j] < '-':            # if the adjacent cell hasn't been converted already
                    q.append(j)           #  append it's index to the queue

            if c > '=':
                z[i] = '│'                # replace a '|' with a '│'

            elif c > '+':
                z[i] = '─═'[a[i]>'-']     # replace a '-' or '=' with '─' or '═' respectively 

            else:                                      # it's a '+'
                U = "┌╒┐╕┬╤@@└╘┘╛┴╧##├╞┤╡┼╪$$"         # even indexes are single horizontal line, 
                                                       # double horizontal lines are at odd indexes

                z[i] = U['='in a[abs(i-1):i+2]::2][k]  # '='in a[abs(i-1):i+2] is true if there is an '=' to the left or right
                                                       # so this selects the odd chars from U
                                                       #  then [k] selects the correct char

 return''.join(z)