How can I plot my financial indicator below a trading chart (produced by TradingChart) with the same x-axis?

It appears you can define your own indicators to be used with TradingChart (this doesn't seem to be documented though):

myIndicator[data_, OptionsPattern[]] := Module[
  {myind},
  myind = 
   Accumulate /@ 
    SplitBy[Differences[Log[QuantityMagnitude[data[[4]]["Values"]]]], Sign] // 
     Flatten;
  {TimeSeries[myind, {Most[data[[4]]["Dates"]]}]}
 ]

This is essentially the code from your question put into a function, with two changes: First, we accept options via OptionsPattern[] (only needed when using something like "IndicatorLayout"->"Overlay", or custom options, supplied via FinancialIndicator[myIndicator, opts]). Second, we get a list 5 TimeSeries objects as data instead of one TimeSeries with 5 entries, so we need to extract the 4th value differently.

Now, to use it, simply wrap the function in FinancialIndicator:

ohlcv = FinancialData["SPY", "OHLCV", {DatePlus[Today, -300], Yesterday}];
TradingChart[ohlcv, {"Volume", FinancialIndicator[myIndicator]}]

We can also add it as an overlay to the main plot:

As you can see, this doesn't make a lot of sense here since the y-values are very different (note the almost flat blue line at the bottom)

Update

Here's an approach to get something similar to the plot in your updated question:

TradingChart[
  ohlcv, {
   "Volume", "BollingerBands",
   FinancialIndicator[
    QuantityMagnitude[{awesomeOscillator, awesomeOscillatormv}, "Dollars"] &,
    "Filling" -> {1 -> 0},
    "RenderingElement" -> {"Dot", "Line"},
    "LineStyle" -> ColorData[91][1]]
   },
  AspectRatio -> 1
  ] /. Inset[pre___, HoldPattern@{w_, h : System`TradingChartDump`$aspectratio 100}] :> Inset[pre, {w, h/4}]

Some notes:

• In this example, I have used a financial indicator of the form FinancialIndicator[data&, ...], where the indicator "function" is an explicit Function expression that simply returns the desired data. As you can see, the name in the indicator label is a bit ugly this way, you can simply use the method with a user-defined function if you want to change it.
• The two data sets are plotted together by grouping them into a single FinancialIndicator
• The styles "RenderingElement" and "LineStyle" can be specified separately per line, the setting for "Filling" works as documented for Filling.
• As you can see, the filling is not a perfect match of the example in your question. This is due to two things: TradingChart seems to resample the data, so the data points are regularly spaced. The filling is generated from a ListLinePlot, so there's no way to get the per-point filling this way.
• The /. Inset[...] part is used to change the aspect ratio of the main plot in relation to the indicator plots. It works by identifying all the insets responsible for the main plot, and dividing their height by 4 (which is the default ration between the heights) to make them all the same height.

Start:

ohlcv = FinancialData["SPY", 
   "OHLCV", {DatePlus[Today, -300], Yesterday}];
chart1 = TradingChart[ohlcv, {"Volume", "BollingerBands"}, 
   ImageSize -> {700, 500}];

---

myind = Accumulate /@ 
    SplitBy[Differences[
      Log[QuantityMagnitude[ohlcv["Values"][[All, 4]]]]], Sign] // 
   Flatten;
myindts = TimeSeries[myind, {Most[ohlcv["Dates"]]}];
chart2 = DateListPlot[myindts, 
  Frame -> {{False, True}, {True, False}}, 
  GridLines -> {{{2020, 1}, {2020, 2}, {2020, 3}, {2020, 4}, {2020, 
      5}, {2020, 6}, {2020, 7}, {2020, 8}, {2020, 9}, {2020, 
      10}}, {-.15, -.10, -0.05, 0, 0.05, 0.10, 0.15}}, 
  GridLinesStyle -> Directive[Lighter@Gray], AspectRatio -> 135/769, 
  ImageSize -> {769, 135}(*,DateTicksFormat\[Rule]{"Month"}*), 
  FrameTicks -> {{{2020, 1}, {2020, 2}, {2020, 3}, {2020, 4}, {2020, 
      5}, {2020, 6}, {2020, 7}, {2020, 8}, {2020, 9}}, {-0.15, -.1, 
     0, .1, 0.15}}, ImagePadding -> {{0, 94}, {25, 0}}, 
  PlotRangePadding -> None, 
  PlotRange -> {{First@Most[ohlcv["Dates"]], 
     Last@Most[ohlcv["Dates"]]}, {-0.175, 0.175}}]

DateListPlot adds a zone at the start and end of the TimeSeries per default.

Column[{chart1, chart2}, Left]

DateListPlot blocks a lot of Options that need to be modified together. The top level problem is match the ImageSize. This only works in combination with AspectRatio due to presettings of DateListPlot.

Next problem is Ticks and Tickslabels.

Next problem is getting the axis from the left to the right for general TimeSeries this works, but the one of the question is trickier.

Seems that the TimeSeries either has different date on pupose or accident. It is easily possible to extend the TimeSeries to the time inteval under purpose in this fashion but the DateListPlot given is the fashion.

My remaining problem is the PlotRangePadding. I have no example or idea on how to do this in a DateListPlot. But I will work on that.

Some further improvement is this:

values = {200, 225, 250, 275, 300, 325, 350};dateslist1 = Prepend[{2020, #} & /@ Range[1, 9], {2019, 12}];
dateslist15 = Append[#, 15] & /@ dateslist;
dateslist = Riffle[dateslist1, dateslist15];
month = Table[AbsoluteTime@dateslist[[i]], {i, 1, Length@dateslist}];
chart3 = TradingChart[
  ohlcv, {"BollingerBands", FinancialIndicator@AccLogDiffs}, 
  PlotRange -> All, 
  FrameTicks -> {{{All, Automatic}, {Automatic, 
      Join[{-.15, 0, .15}, values]}}}, 
  GridLines -> {Automatic, Join[{-.15, 0, .15, .2}, values]}, 
  DisplayFunction -> Identity, ImageSize -> Large]

ohlcv1 = FinancialData["SPY", 
   "OHLCV", {DatePlus[Today, -300], DatePlus[Today, -265]}];
chart3 = TradingChart[
  ohlcv1, {"BollingerBands", FinancialIndicator@AccLogDiffs}, 
  PlotRange -> All, AspectRatio -> 580/350, ImageSize -> {580, 350}, 
  FrameTicks -> {{None, Automatic}, {None, None}}, 
  GridLines -> {Automatic, Join[{-.15, 0, .15, .2}, values]}, 
  DisplayFunction -> Identity]

The time series length is of great importance to TradingChart. It is is spread over two years as in the question the "Month" is internally replace with "Year" and that is very hard to change. It does take GridLines, but not FrameTicks. The change is "Year" and "Month" is really great and instructive.

Thanks to the answer Extracting financial indicator data from trading chart of @kuba we get further to

chart1 = TradingChart[ohlcv, {"Volume", "BollingerBands"}, 
   Appearance -> "Line", PerformanceGoal :> "Speed"];    
Cases[chart2, _Graphics, Infinity][[5]]

So for the Options Appearance and Performance with the shown methods gives the conformation the process is the correct way. All that is in need to be done place the fifth part of it with the axis You will like to have.

The interactive cursor is a problem but this can be added the same way as this in a second step:

ohlcv = FinancialData["SPY", 
   "OHLCV", {DatePlus[Today, -300], Yesterday}];
chart11 = TradingChart[ohlcv, {"Volume", "BollingerBands"}, 
   Appearance -> "Line", PerformanceGoal :> "Speed"];

---

dateslist=myindts[[2, 11, 2, 2, 1, 2, 1, 2]];
monthsmy=myindts[[2, 11, 2, 2, 1, 2, 2, 2]];
labels = Partition[
  Riffle[monthsmy[[1]], (12.5 + 
      185.5 (%289 - Min[dateslist])/((Max[dateslist] - Min[dateslist])))[[1]]], 2]

12.5 is the offset of the position of `19 and 185.5 is an estimate for the position of the Sep in the new graph.

With this, we can define a replacement rule for the Graphics displaying the unwanted Years.

InputForm@(Cases[chart1, _Graphics, \[Infinity]][[5]])[[1, 6, 1, 3, 1, 2, 1, 3, 1, 1 ;; 2]] /. 
 listts : {_Text, _Text} :> (texts /@ labels)

Plug it into the inactive TradingChart:

chart11 /. listts : {_Text, _Text} :> (texts /@ labels)